% Mizar problem: e4_6,arytm_3,106,36 
include('Axioms/SET010^0.ax').
thf(k11_ordinal2_type,type,( k11_ordinal2: $i > $i > $i )).
thf(k1_xboole_0_type,type,( k1_xboole_0: $i )).
thf(k4_ordinal1_type,type,( k4_ordinal1: $i )).
thf(k9_ordinal3_type,type,( k9_ordinal3: $i > $i > $i )).
thf(r1_arytm_3_type,type,( r1_arytm_3: $i > $i > $o )).
thf(r1_ordinal1_type,type,( r1_ordinal1: $i > $i > $o )).
thf(r1_tarski_type,type,( r1_tarski: $i > $i > $o )).
thf(v1_ordinal1_type,type,( v1_ordinal1: $i > $o )).
thf(v1_xboole_0_type,type,( v1_xboole_0: $i > $o )).
thf(v2_ordinal1_type,type,( v2_ordinal1: $i > $o )).
thf(v3_ordinal1_type,type,( v3_ordinal1: $i > $o )).
thf(v5_ordinal1_type,type,( v5_ordinal1: $i > $o )).
thf(v6_ordinal1_type,type,( v6_ordinal1: $i > $o )).
thf(v7_ordinal1_type,type,( v7_ordinal1: $i > $o )).
thf(dt_k11_ordinal2,axiom,( ! [A: $i] : ! [B: $i] : ( ( ( v3_ordinal1 @ A ) & ( v3_ordinal1 @ B ) ) => ( v3_ordinal1 @ ( k11_ordinal2 @ A @ B ) ) ) ),file(ordinal2,k11_ordinal2)).
thf(cc2_ordinal1,axiom,( ! [A: $i] : ( ( ( v1_ordinal1 @ A ) & ( v2_ordinal1 @ A ) ) => ( v3_ordinal1 @ A ) ) ),file(ordinal1,cc2_ordinal1)).
thf(cc3_ordinal1,axiom,( ! [A: $i] : ( ( v1_xboole_0 @ A ) => ( v3_ordinal1 @ A ) ) ),file(ordinal1,cc3_ordinal1)).
thf(cc4_ordinal1,axiom,( ! [A: $i] : ( ( v1_xboole_0 @ A ) => ( v5_ordinal1 @ A ) ) ),file(ordinal1,cc4_ordinal1)).
thf(cc7_ordinal1,axiom,( ! [A: $i] : ( ( v1_xboole_0 @ A ) => ( v7_ordinal1 @ A ) ) ),file(ordinal1,cc7_ordinal1)).
thf(cc9_ordinal1,axiom,( ! [A: $i] : ( ( v1_xboole_0 @ A ) => ( v6_ordinal1 @ A ) ) ),file(ordinal1,cc9_ordinal1)).
thf(fc4_ordinal3,axiom,( ! [A: $i] : ! [B: $i] : ( ( ( v3_ordinal1 @ A ) & ( v7_ordinal1 @ A ) & ( v3_ordinal1 @ B ) & ( v7_ordinal1 @ B ) ) => ( ( v3_ordinal1 @ ( k11_ordinal2 @ A @ B ) ) & ( v7_ordinal1 @ ( k11_ordinal2 @ A @ B ) ) ) ) ),file(ordinal3,fc4_ordinal3)).
thf(rc1_xboole_0,axiom,( ? [A: $i] : ( v1_xboole_0 @ A ) ),file(xboole_0,rc1_xboole_0)).
thf(rc2_ordinal1,axiom,( ? [A: $i] : ( ~ ( v1_xboole_0 @ A ) & ( v1_ordinal1 @ A ) & ( v2_ordinal1 @ A ) & ( v3_ordinal1 @ A ) ) ),file(ordinal1,rc2_ordinal1)).
thf(rc2_xboole_0,axiom,( ? [A: $i] : ~ ( v1_xboole_0 @ A ) ),file(xboole_0,rc2_xboole_0)).
thf(rc5_ordinal1,axiom,( ? [A: $i] : ( ~ ( v1_xboole_0 @ A ) & ( v7_ordinal1 @ A ) ) ),file(ordinal1,rc5_ordinal1)).
thf(commutativity_k9_ordinal3,axiom,( ! [A: $i] : ! [B: $i] : ( ( ( v3_ordinal1 @ A ) & ( v7_ordinal1 @ A ) & ( v3_ordinal1 @ B ) & ( v7_ordinal1 @ B ) ) => ( = @ ( k9_ordinal3 @ A @ B ) @ ( k9_ordinal3 @ B @ A ) ) ) ),file(ordinal3,k9_ordinal3)).
thf(symmetry_r1_arytm_3,axiom,( ! [A: $i] : ! [B: $i] : ( ( ( v3_ordinal1 @ A ) & ( v3_ordinal1 @ B ) ) => ( ( r1_arytm_3 @ A @ B ) => ( r1_arytm_3 @ B @ A ) ) ) ),file(arytm_3,r1_arytm_3)).
thf(reflexivity_r1_ordinal1,axiom,( ! [A: $i] : ! [B: $i] : ( ( ( v3_ordinal1 @ A ) & ( v3_ordinal1 @ B ) ) => ( r1_ordinal1 @ A @ A ) ) ),file(ordinal1,r1_ordinal1)).
thf(connectedness_r1_ordinal1,axiom,( ! [A: $i] : ! [B: $i] : ( ( ( v3_ordinal1 @ A ) & ( v3_ordinal1 @ B ) ) => ( ( r1_ordinal1 @ A @ B ) | ( r1_ordinal1 @ B @ A ) ) ) ),file(ordinal1,r1_ordinal1)).
thf(reflexivity_r1_tarski,axiom,( ! [A: $i] : ! [B: $i] : ( r1_tarski @ A @ A ) ),file(tarski,r1_tarski)).
thf(antisymmetry_r2_hidden,axiom,( ! [A: $i] : ! [B: $i] : ( ( r2_hidden @ A @ B ) => ~ ( r2_hidden @ B @ A ) ) ),file(hidden,r2_hidden)).
thf(redefinition_k9_ordinal3,axiom,( ! [A: $i] : ! [B: $i] : ( ( ( v3_ordinal1 @ A ) & ( v7_ordinal1 @ A ) & ( v3_ordinal1 @ B ) & ( v7_ordinal1 @ B ) ) => ( = @ ( k9_ordinal3 @ A @ B ) @ ( k11_ordinal2 @ A @ B ) ) ) ),file(ordinal3,k9_ordinal3)).
thf(redefinition_r1_ordinal1,axiom,( ! [A: $i] : ! [B: $i] : ( ( ( v3_ordinal1 @ A ) & ( v3_ordinal1 @ B ) ) => ( ( r1_ordinal1 @ A @ B ) <=> ( r1_tarski @ A @ B ) ) ) ),file(ordinal1,r1_ordinal1)).
thf(dt_k1_xboole_0,axiom,( $true ),file(xboole_0,k1_xboole_0)).
thf(dt_k4_ordinal1,axiom,( $true ),file(ordinal1,k4_ordinal1)).
thf(dt_k9_ordinal3,axiom,( ! [A: $i] : ! [B: $i] : ( ( ( v3_ordinal1 @ A ) & ( v7_ordinal1 @ A ) & ( v3_ordinal1 @ B ) & ( v7_ordinal1 @ B ) ) => ( v3_ordinal1 @ ( k9_ordinal3 @ A @ B ) ) ) ),file(ordinal3,k9_ordinal3)).
thf(cc1_ordinal1,axiom,( ! [A: $i] : ( ( v3_ordinal1 @ A ) => ( ( v1_ordinal1 @ A ) & ( v2_ordinal1 @ A ) ) ) ),file(ordinal1,cc1_ordinal1)).
thf(cc6_ordinal1,axiom,( ! [A: $i] : ( ( v7_ordinal1 @ A ) => ( v3_ordinal1 @ A ) ) ),file(ordinal1,cc6_ordinal1)).
thf(fc1_xboole_0,axiom,( v1_xboole_0 @ k1_xboole_0 ),file(xboole_0,fc1_xboole_0)).
thf(fc6_ordinal1,axiom,( ~ ( v1_xboole_0 @ k4_ordinal1 ) & ( v3_ordinal1 @ k4_ordinal1 ) ),file(ordinal1,fc6_ordinal1)).
thf(rc1_ordinal1,axiom,( ? [A: $i] : ( v3_ordinal1 @ A ) ),file(ordinal1,rc1_ordinal1)).
thf(rc4_ordinal1,axiom,( ? [A: $i] : ( v7_ordinal1 @ A ) ),file(ordinal1,rc4_ordinal1)).
thf(e3_6__arytm_3,axiom,( ? [A: $i] : ( ( v3_ordinal1 @ A ) & ? [B: $i] : ( ( v3_ordinal1 @ B ) & ( r1_tarski @ B @ A ) & ( r2_hidden @ A @ k4_ordinal1 ) & ~ ( = @ A @ k1_xboole_0 ) & ! [C: $i] : ( ( ( v3_ordinal1 @ C ) & ( v7_ordinal1 @ C ) ) => ! [D: $i] : ( ( ( v3_ordinal1 @ D ) & ( v7_ordinal1 @ D ) ) => ! [E: $i] : ( ( ( v3_ordinal1 @ E ) & ( v7_ordinal1 @ E ) ) => ~ ( ( r1_arytm_3 @ D @ E ) & ( = @ A @ ( k9_ordinal3 @ C @ D ) ) & ( = @ B @ ( k9_ordinal3 @ C @ E ) ) ) ) ) ) ) ) ),file(arytm_3,e3_6__arytm_3)).
thf(s1_ordinal1,axiom,( ! [A: $i > $o] : ( ? [B: $i] : ( ( v3_ordinal1 @ B ) & ( A @ B ) ) => ? [B: $i] : ( ( v3_ordinal1 @ B ) & ( A @ B ) & ! [C: $i] : ( ( v3_ordinal1 @ C ) => ( ( A @ C ) => ( r1_ordinal1 @ B @ C ) ) ) ) ) ),file(ordinal1,s1_ordinal1)).
thf(e4_6__arytm_3,conjecture,( ? [A: $i] : ( ( v3_ordinal1 @ A ) & ? [B: $i] : ( ( v3_ordinal1 @ B ) & ( r1_tarski @ B @ A ) & ( r2_hidden @ A @ k4_ordinal1 ) & ~ ( = @ A @ k1_xboole_0 ) & ! [C: $i] : ( ( ( v3_ordinal1 @ C ) & ( v7_ordinal1 @ C ) ) => ! [D: $i] : ( ( ( v3_ordinal1 @ D ) & ( v7_ordinal1 @ D ) ) => ! [E: $i] : ( ( ( v3_ordinal1 @ E ) & ( v7_ordinal1 @ E ) ) => ~ ( ( r1_arytm_3 @ D @ E ) & ( = @ A @ ( k9_ordinal3 @ C @ D ) ) & ( = @ B @ ( k9_ordinal3 @ C @ E ) ) ) ) ) ) ) & ! [B: $i] : ( ( v3_ordinal1 @ B ) => ( ? [C: $i] : ( ( v3_ordinal1 @ C ) & ( r1_tarski @ C @ B ) & ( r2_hidden @ B @ k4_ordinal1 ) & ~ ( = @ B @ k1_xboole_0 ) & ! [D: $i] : ( ( ( v3_ordinal1 @ D ) & ( v7_ordinal1 @ D ) ) => ! [E: $i] : ( ( ( v3_ordinal1 @ E ) & ( v7_ordinal1 @ E ) ) => ! [F: $i] : ( ( ( v3_ordinal1 @ F ) & ( v7_ordinal1 @ F ) ) => ~ ( ( r1_arytm_3 @ E @ F ) & ( = @ B @ ( k9_ordinal3 @ D @ E ) ) & ( = @ C @ ( k9_ordinal3 @ D @ F ) ) ) ) ) ) ) => ( r1_ordinal1 @ A @ B ) ) ) ) ),file(arytm_3,e4_6__arytm_3)).
