% Mizar problem: e2_18_1,fraenkel,416,6 
include('Axioms/SET010^2.ax').
thf(f1_s15_fraenkel_type,type,( f1_s15_fraenkel: $i )).
thf(f2_s15_fraenkel_type,type,( f2_s15_fraenkel: $i )).
thf(f3_s15_fraenkel_type,type,( f3_s15_fraenkel: $i > $i > $i )).
thf(m1_subset_1_type,type,( m1_subset_1: $i > $i > $o )).
thf(p1_s15_fraenkel_type,type,( p1_s15_fraenkel: $i > $i > $o )).
thf(p2_s15_fraenkel_type,type,( p2_s15_fraenkel: $i > $o )).
thf(antisymmetry_r2_hidden,axiom,( ! [A: $i] : ! [B: $i] : ( ( r2_hidden @ A @ B ) => ~ ( r2_hidden @ B @ A ) ) ),file(hidden,r2_hidden)).
thf(dt_m1_subset_1,axiom,( $true ),file(subset_1,m1_subset_1)).
thf(dt_f1_s15_fraenkel,axiom,( $true ),file(fraenkel,f1_s15_fraenkel)).
thf(dt_f2_s15_fraenkel,axiom,( $true ),file(fraenkel,f2_s15_fraenkel)).
thf(dt_f3_s15_fraenkel,axiom,( $true ),file(fraenkel,f3_s15_fraenkel)).
thf(e1_18_1__fraenkel,axiom,( ! [A: $i] : ( ( m1_subset_1 @ A @ f1_s15_fraenkel ) => ! [B: $i] : ( ( m1_subset_1 @ B @ f2_s15_fraenkel ) => ( ( ( r2_hidden @ A @ ( replSep1 @ ^ [C: $i] : ( m1_subset_1 @ C @ f1_s15_fraenkel ) @ ^ [C: $i] : C @ ^ [C: $i] : ( p2_s15_fraenkel @ C ) ) ) & ( p1_s15_fraenkel @ A @ B ) ) <=> ( ( p2_s15_fraenkel @ A ) & ( p1_s15_fraenkel @ A @ B ) ) ) ) ) ),file(fraenkel,e1_18_1__fraenkel)).
thf(s4_fraenkel,axiom,( ! [A: $i > $i > $i] : ! [B: $i > $i > $o] : ! [C: $i > $i > $o] : ! [D: $i] : ! [E: $i] : ( ! [F: $i] : ( ( m1_subset_1 @ F @ E ) => ! [G: $i] : ( ( m1_subset_1 @ G @ D ) => ( ( C @ F @ G ) <=> ( B @ F @ G ) ) ) ) => ( = @ ( replSep2 @ ^ [F: $i] : ( m1_subset_1 @ F @ E ) @ ^ [F: $i] : ^ [G: $i] : ( m1_subset_1 @ G @ D ) @ ^ [F: $i] : ^ [G: $i] : ( A @ F @ G ) @ ^ [F: $i] : ^ [G: $i] : ( C @ F @ G ) ) @ ( replSep2 @ ^ [F: $i] : ( m1_subset_1 @ F @ E ) @ ^ [F: $i] : ^ [G: $i] : ( m1_subset_1 @ G @ D ) @ ^ [F: $i] : ^ [G: $i] : ( A @ F @ G ) @ ^ [F: $i] : ^ [G: $i] : ( B @ F @ G ) ) ) ) ),file(fraenkel,s4_fraenkel)).
thf(e2_18_1__fraenkel,conjecture,( = @ ( replSep2 @ ^ [A: $i] : ( m1_subset_1 @ A @ f1_s15_fraenkel ) @ ^ [A: $i] : ^ [B: $i] : ( m1_subset_1 @ B @ f2_s15_fraenkel ) @ ^ [A: $i] : ^ [B: $i] : ( f3_s15_fraenkel @ A @ B ) @ ^ [A: $i] : ^ [B: $i] : ( ( r2_hidden @ A @ ( replSep1 @ ^ [C: $i] : ( m1_subset_1 @ C @ f1_s15_fraenkel ) @ ^ [C: $i] : C @ ^ [C: $i] : ( p2_s15_fraenkel @ C ) ) ) & ( p1_s15_fraenkel @ A @ B ) ) ) @ ( replSep2 @ ^ [A: $i] : ( m1_subset_1 @ A @ f1_s15_fraenkel ) @ ^ [A: $i] : ^ [B: $i] : ( m1_subset_1 @ B @ f2_s15_fraenkel ) @ ^ [A: $i] : ^ [B: $i] : ( f3_s15_fraenkel @ A @ B ) @ ^ [A: $i] : ^ [B: $i] : ( ( p2_s15_fraenkel @ A ) & ( p1_s15_fraenkel @ A @ B ) ) ) ),file(fraenkel,e2_18_1__fraenkel)).
thf(sethood_m1_subset_1,axiom,( ! [A: $i] : ( sethood @ ^ [X: $i] : ( m1_subset_1 @ X @ A ) ) ),file(subset_1,m1_subset_1)).
