% Mizar problem: e4_41,domain_1,698,17 
include('Axioms/SET010^1.ax').
thf(c1_41_type,type,( c1_41__domain_1: $i )).
thf(c2_41_type,type,( c2_41__domain_1: $i )).
thf(c3_41_type,type,( c3_41__domain_1: $i )).
thf(k1_xboole_0_type,type,( k1_xboole_0: $i )).
thf(k1_zfmisc_1_type,type,( k1_zfmisc_1: $i > $i )).
thf(k2_xboole_0_type,type,( k2_xboole_0: $i > $i > $i )).
thf(k4_xboole_0_type,type,( k4_xboole_0: $i > $i > $i )).
thf(k5_subset_1_type,type,( k5_subset_1: $i > $i > $i > $i )).
thf(k5_xboole_0_type,type,( k5_xboole_0: $i > $i > $i )).
thf(m1_subset_1_type,type,( m1_subset_1: $i > $i > $o )).
thf(r1_tarski_type,type,( r1_tarski: $i > $i > $o )).
thf(v1_relat_1_type,type,( v1_relat_1: $i > $o )).
thf(v1_subset_1_type,type,( v1_subset_1: $i > $i > $o )).
thf(v1_xboole_0_type,type,( v1_xboole_0: $i > $o )).
thf(v2_relat_1_type,type,( v2_relat_1: $i > $o )).
thf(v3_relat_1_type,type,( v3_relat_1: $i > $o )).
thf(commutativity_k2_xboole_0,axiom,( ! [A: $i] : ! [B: $i] : ( = @ ( k2_xboole_0 @ A @ B ) @ ( k2_xboole_0 @ B @ A ) ) ),file(xboole_0,k2_xboole_0)).
thf(idempotence_k2_xboole_0,axiom,( ! [A: $i] : ! [B: $i] : ( = @ ( k2_xboole_0 @ A @ A ) @ A ) ),file(xboole_0,k2_xboole_0)).
thf(reflexivity_r1_tarski,axiom,( ! [A: $i] : ! [B: $i] : ( r1_tarski @ A @ A ) ),file(tarski,r1_tarski)).
thf(dt_k2_xboole_0,axiom,( $true ),file(xboole_0,k2_xboole_0)).
thf(dt_k4_xboole_0,axiom,( $true ),file(xboole_0,k4_xboole_0)).
thf(cc2_subset_1,axiom,( ! [A: $i] : ( ~ ( v1_xboole_0 @ A ) => ! [B: $i] : ( ( m1_subset_1 @ B @ ( k1_zfmisc_1 @ A ) ) => ( ~ ( v1_subset_1 @ B @ A ) => ~ ( v1_xboole_0 @ B ) ) ) ) ),file(subset_1,cc2_subset_1)).
thf(fc2_relat_1,axiom,( ! [A: $i] : ! [B: $i] : ( ( v1_relat_1 @ A ) => ( v1_relat_1 @ ( k4_xboole_0 @ A @ B ) ) ) ),file(relat_1,fc2_relat_1)).
thf(fc3_relat_1,axiom,( ! [A: $i] : ! [B: $i] : ( ( ( v1_relat_1 @ A ) & ( v1_relat_1 @ B ) ) => ( v1_relat_1 @ ( k2_xboole_0 @ A @ B ) ) ) ),file(relat_1,fc3_relat_1)).
thf(fc4_xboole_0,axiom,( ! [A: $i] : ! [B: $i] : ( ~ ( v1_xboole_0 @ A ) => ~ ( v1_xboole_0 @ ( k2_xboole_0 @ A @ B ) ) ) ),file(xboole_0,fc4_xboole_0)).
thf(fc5_xboole_0,axiom,( ! [A: $i] : ! [B: $i] : ( ~ ( v1_xboole_0 @ A ) => ~ ( v1_xboole_0 @ ( k2_xboole_0 @ B @ A ) ) ) ),file(xboole_0,fc5_xboole_0)).
thf(rc2_relat_1,axiom,( ? [A: $i] : ( ( v1_relat_1 @ A ) & ( v2_relat_1 @ A ) ) ),file(relat_1,rc2_relat_1)).
thf(rc3_subset_1,axiom,( ! [A: $i] : ? [B: $i] : ( ( m1_subset_1 @ B @ ( k1_zfmisc_1 @ A ) ) & ~ ( v1_subset_1 @ B @ A ) ) ),file(subset_1,rc3_subset_1)).
thf(rc4_subset_1,axiom,( ! [A: $i] : ( ~ ( v1_xboole_0 @ A ) => ? [B: $i] : ( ( m1_subset_1 @ B @ ( k1_zfmisc_1 @ A ) ) & ( v1_subset_1 @ B @ A ) ) ) ),file(subset_1,rc4_subset_1)).
thf(t1_boole,axiom,( ! [A: $i] : ( = @ ( k2_xboole_0 @ A @ k1_xboole_0 ) @ A ) ),file(boole,t1_boole)).
thf(t3_boole,axiom,( ! [A: $i] : ( = @ ( k4_xboole_0 @ A @ k1_xboole_0 ) @ A ) ),file(boole,t3_boole)).
thf(t4_boole,axiom,( ! [A: $i] : ( = @ ( k4_xboole_0 @ k1_xboole_0 @ A ) @ k1_xboole_0 ) ),file(boole,t4_boole)).
thf(commutativity_k5_xboole_0,axiom,( ! [A: $i] : ! [B: $i] : ( = @ ( k5_xboole_0 @ A @ B ) @ ( k5_xboole_0 @ B @ A ) ) ),file(xboole_0,k5_xboole_0)).
thf(dt_k1_xboole_0,axiom,( $true ),file(xboole_0,k1_xboole_0)).
thf(dt_k1_zfmisc_1,axiom,( $true ),file(zfmisc_1,k1_zfmisc_1)).
thf(dt_k5_xboole_0,axiom,( $true ),file(xboole_0,k5_xboole_0)).
thf(cc1_subset_1,axiom,( ! [A: $i] : ( ( v1_xboole_0 @ A ) => ! [B: $i] : ( ( m1_subset_1 @ B @ ( k1_zfmisc_1 @ A ) ) => ( v1_xboole_0 @ B ) ) ) ),file(subset_1,cc1_subset_1)).
thf(cc2_relat_1,axiom,( ! [A: $i] : ( ( v1_relat_1 @ A ) => ! [B: $i] : ( ( m1_subset_1 @ B @ ( k1_zfmisc_1 @ A ) ) => ( v1_relat_1 @ B ) ) ) ),file(relat_1,cc2_relat_1)).
thf(cc3_relat_1,axiom,( ! [A: $i] : ( ( ( v1_xboole_0 @ A ) & ( v1_relat_1 @ A ) ) => ( ( v1_relat_1 @ A ) & ( v3_relat_1 @ A ) ) ) ),file(relat_1,cc3_relat_1)).
thf(cc3_subset_1,axiom,( ! [A: $i] : ( ~ ( v1_xboole_0 @ A ) => ! [B: $i] : ( ( m1_subset_1 @ B @ ( k1_zfmisc_1 @ A ) ) => ( ( v1_xboole_0 @ B ) => ( v1_subset_1 @ B @ A ) ) ) ) ),file(subset_1,cc3_subset_1)).
thf(cc4_relat_1,axiom,( ! [A: $i] : ( ( ( v1_xboole_0 @ A ) & ( v1_relat_1 @ A ) ) => ( ( v1_relat_1 @ A ) & ( v2_relat_1 @ A ) ) ) ),file(relat_1,cc4_relat_1)).
thf(cc4_subset_1,axiom,( ! [A: $i] : ( ( v1_xboole_0 @ A ) => ! [B: $i] : ( ( m1_subset_1 @ B @ ( k1_zfmisc_1 @ A ) ) => ~ ( v1_subset_1 @ B @ A ) ) ) ),file(subset_1,cc4_subset_1)).
thf(fc1_subset_1,axiom,( ! [A: $i] : ~ ( v1_xboole_0 @ ( k1_zfmisc_1 @ A ) ) ),file(subset_1,fc1_subset_1)).
thf(fc1_xboole_0,axiom,( v1_xboole_0 @ k1_xboole_0 ),file(xboole_0,fc1_xboole_0)).
thf(fc4_relat_1,axiom,( ! [A: $i] : ! [B: $i] : ( ( ( v1_relat_1 @ A ) & ( v1_relat_1 @ B ) ) => ( v1_relat_1 @ ( k5_xboole_0 @ A @ B ) ) ) ),file(relat_1,fc4_relat_1)).
thf(rc1_relat_1,axiom,( ? [A: $i] : ( ~ ( v1_xboole_0 @ A ) & ( v1_relat_1 @ A ) ) ),file(relat_1,rc1_relat_1)).
thf(rc1_subset_1,axiom,( ! [A: $i] : ( ~ ( v1_xboole_0 @ A ) => ? [B: $i] : ( ( m1_subset_1 @ B @ ( k1_zfmisc_1 @ A ) ) & ~ ( v1_xboole_0 @ B ) ) ) ),file(subset_1,rc1_subset_1)).
thf(rc2_subset_1,axiom,( ! [A: $i] : ? [B: $i] : ( ( m1_subset_1 @ B @ ( k1_zfmisc_1 @ A ) ) & ( v1_xboole_0 @ B ) ) ),file(subset_1,rc2_subset_1)).
thf(t3_subset,axiom,( ! [A: $i] : ! [B: $i] : ( ( m1_subset_1 @ A @ ( k1_zfmisc_1 @ B ) ) <=> ( r1_tarski @ A @ B ) ) ),file(subset,t3_subset)).
thf(t4_subset,axiom,( ! [A: $i] : ! [B: $i] : ! [C: $i] : ( ( ( r2_hidden @ A @ B ) & ( m1_subset_1 @ B @ ( k1_zfmisc_1 @ C ) ) ) => ( m1_subset_1 @ A @ C ) ) ),file(subset,t4_subset)).
thf(t5_boole,axiom,( ! [A: $i] : ( = @ ( k5_xboole_0 @ A @ k1_xboole_0 ) @ A ) ),file(boole,t5_boole)).
thf(t5_subset,axiom,( ! [A: $i] : ! [B: $i] : ! [C: $i] : ~ ( ( r2_hidden @ A @ B ) & ( m1_subset_1 @ B @ ( k1_zfmisc_1 @ C ) ) & ( v1_xboole_0 @ C ) ) ),file(subset,t5_subset)).
thf(d2_xboole_0,axiom,( = @ k1_xboole_0 @ ( eps @ ^ [A: $i] : ( v1_xboole_0 @ A ) ) ),file(xboole_0,d2_xboole_0)).
thf(d6_xboole_0,axiom,( ! [A: $i] : ! [B: $i] : ( = @ ( k5_xboole_0 @ A @ B ) @ ( k2_xboole_0 @ ( k4_xboole_0 @ A @ B ) @ ( k4_xboole_0 @ B @ A ) ) ) ),file(xboole_0,d6_xboole_0)).
thf(commutativity_k5_subset_1,axiom,( ! [A: $i] : ! [B: $i] : ! [C: $i] : ( ( ( m1_subset_1 @ B @ ( k1_zfmisc_1 @ A ) ) & ( m1_subset_1 @ C @ ( k1_zfmisc_1 @ A ) ) ) => ( = @ ( k5_subset_1 @ A @ B @ C ) @ ( k5_subset_1 @ A @ C @ B ) ) ) ),file(subset_1,k5_subset_1)).
thf(antisymmetry_r2_hidden,axiom,( ! [A: $i] : ! [B: $i] : ( ( r2_hidden @ A @ B ) => ~ ( r2_hidden @ B @ A ) ) ),file(hidden,r2_hidden)).
thf(existence_m1_subset_1,axiom,( ! [A: $i] : ? [B: $i] : ( m1_subset_1 @ B @ A ) ),file(subset_1,m1_subset_1)).
thf(redefinition_k5_subset_1,axiom,( ! [A: $i] : ! [B: $i] : ! [C: $i] : ( ( ( m1_subset_1 @ B @ ( k1_zfmisc_1 @ A ) ) & ( m1_subset_1 @ C @ ( k1_zfmisc_1 @ A ) ) ) => ( = @ ( k5_subset_1 @ A @ B @ C ) @ ( k5_xboole_0 @ B @ C ) ) ) ),file(subset_1,k5_subset_1)).
thf(dt_k5_subset_1,axiom,( ! [A: $i] : ! [B: $i] : ! [C: $i] : ( ( ( m1_subset_1 @ B @ ( k1_zfmisc_1 @ A ) ) & ( m1_subset_1 @ C @ ( k1_zfmisc_1 @ A ) ) ) => ( m1_subset_1 @ ( k5_subset_1 @ A @ B @ C ) @ ( k1_zfmisc_1 @ A ) ) ) ),file(subset_1,k5_subset_1)).
thf(dt_m1_subset_1,axiom,( $true ),file(subset_1,m1_subset_1)).
thf(dt_c1_41__domain_1,axiom,( ~ ( v1_xboole_0 @ c1_41__domain_1 ) ),file(domain_1,c1_41__domain_1)).
thf(dt_c2_41__domain_1,axiom,( m1_subset_1 @ c2_41__domain_1 @ ( k1_zfmisc_1 @ c1_41__domain_1 ) ),file(domain_1,c2_41__domain_1)).
thf(dt_c3_41__domain_1,axiom,( m1_subset_1 @ c3_41__domain_1 @ ( k1_zfmisc_1 @ c1_41__domain_1 ) ),file(domain_1,c3_41__domain_1)).
thf(cc1_relat_1,axiom,( ! [A: $i] : ( ( v1_xboole_0 @ A ) => ( v1_relat_1 @ A ) ) ),file(relat_1,cc1_relat_1)).
thf(rc1_xboole_0,axiom,( ? [A: $i] : ( v1_xboole_0 @ A ) ),file(xboole_0,rc1_xboole_0)).
thf(rc2_xboole_0,axiom,( ? [A: $i] : ~ ( v1_xboole_0 @ A ) ),file(xboole_0,rc2_xboole_0)).
thf(t1_subset,axiom,( ! [A: $i] : ! [B: $i] : ( ( r2_hidden @ A @ B ) => ( m1_subset_1 @ A @ B ) ) ),file(subset,t1_subset)).
thf(t2_subset,axiom,( ! [A: $i] : ! [B: $i] : ( ( m1_subset_1 @ A @ B ) => ( ( v1_xboole_0 @ B ) | ( r2_hidden @ A @ B ) ) ) ),file(subset,t2_subset)).
thf(t6_boole,axiom,( ! [A: $i] : ( ( v1_xboole_0 @ A ) => ( = @ A @ k1_xboole_0 ) ) ),file(boole,t6_boole)).
thf(t7_boole,axiom,( ! [A: $i] : ! [B: $i] : ~ ( ( r2_hidden @ A @ B ) & ( v1_xboole_0 @ B ) ) ),file(boole,t7_boole)).
thf(t8_boole,axiom,( ! [A: $i] : ! [B: $i] : ~ ( ( v1_xboole_0 @ A ) & ~ ( = @ A @ B ) & ( v1_xboole_0 @ B ) ) ),file(boole,t8_boole)).
thf(e3_41__domain_1,axiom,( ! [A: $i] : ( ~ ( v1_xboole_0 @ A ) => ( ! [B: $i] : ( ( m1_subset_1 @ B @ A ) => ( ( ( ( r2_hidden @ B @ c2_41__domain_1 ) & ~ ( r2_hidden @ B @ c3_41__domain_1 ) ) | ( ~ ( r2_hidden @ B @ c2_41__domain_1 ) & ( r2_hidden @ B @ c3_41__domain_1 ) ) ) <=> ( ~ ( r2_hidden @ B @ c2_41__domain_1 ) <=> ( r2_hidden @ B @ c3_41__domain_1 ) ) ) ) => ( = @ ( replSep1 @ ^ [B: $i] : ( m1_subset_1 @ B @ A ) @ ^ [B: $i] : B @ ^ [B: $i] : ( ( ( r2_hidden @ B @ c2_41__domain_1 ) & ~ ( r2_hidden @ B @ c3_41__domain_1 ) ) | ( ~ ( r2_hidden @ B @ c2_41__domain_1 ) & ( r2_hidden @ B @ c3_41__domain_1 ) ) ) ) @ ( replSep1 @ ^ [B: $i] : ( m1_subset_1 @ B @ A ) @ ^ [B: $i] : B @ ^ [B: $i] : ( ~ ( r2_hidden @ B @ c2_41__domain_1 ) <=> ( r2_hidden @ B @ c3_41__domain_1 ) ) ) ) ) ) ),file(domain_1,e3_41__domain_1)).
thf(e1_41__domain_1,axiom,( ! [A: $i] : ( ( m1_subset_1 @ A @ c1_41__domain_1 ) => ( ( ( ( r2_hidden @ A @ c2_41__domain_1 ) & ~ ( r2_hidden @ A @ c3_41__domain_1 ) ) | ( ~ ( r2_hidden @ A @ c2_41__domain_1 ) & ( r2_hidden @ A @ c3_41__domain_1 ) ) ) <=> ( ~ ( r2_hidden @ A @ c2_41__domain_1 ) <=> ( r2_hidden @ A @ c3_41__domain_1 ) ) ) ) ),file(domain_1,e1_41__domain_1)).
thf(e2_41__domain_1,axiom,( = @ ( k5_subset_1 @ c1_41__domain_1 @ c2_41__domain_1 @ c3_41__domain_1 ) @ ( replSep1 @ ^ [A: $i] : ( m1_subset_1 @ A @ c1_41__domain_1 ) @ ^ [A: $i] : A @ ^ [A: $i] : ( ( ( r2_hidden @ A @ c2_41__domain_1 ) & ~ ( r2_hidden @ A @ c3_41__domain_1 ) ) | ( ~ ( r2_hidden @ A @ c2_41__domain_1 ) & ( r2_hidden @ A @ c3_41__domain_1 ) ) ) ) ),file(domain_1,e2_41__domain_1)).
thf(e4_41__domain_1,conjecture,( = @ ( k5_subset_1 @ c1_41__domain_1 @ c2_41__domain_1 @ c3_41__domain_1 ) @ ( replSep1 @ ^ [A: $i] : ( m1_subset_1 @ A @ c1_41__domain_1 ) @ ^ [A: $i] : A @ ^ [A: $i] : ( ~ ( r2_hidden @ A @ c2_41__domain_1 ) <=> ( r2_hidden @ A @ c3_41__domain_1 ) ) ) ),file(domain_1,e4_41__domain_1)).
thf(sethood_m1_subset_1,axiom,( ! [A: $i] : ( sethood @ ^ [X: $i] : ( m1_subset_1 @ X @ A ) ) ),file(subset_1,m1_subset_1)).
