% Mizar problem: e2_23_1,fraenkel,501,55 
include('Axioms/SET010^2.ax').
thf(f1_s20_fraenkel_type,type,( f1_s20_fraenkel: $i )).
thf(f2_s20_fraenkel_type,type,( f2_s20_fraenkel: $i )).
thf(f3_s20_fraenkel_type,type,( f3_s20_fraenkel: $i > $i > $i )).
thf(f4_s20_fraenkel_type,type,( f4_s20_fraenkel: $i )).
thf(m1_subset_1_type,type,( m1_subset_1: $i > $i > $o )).
thf(p1_s20_fraenkel_type,type,( p1_s20_fraenkel: $i > $i > $o )).
thf(dt_m1_subset_1,axiom,( $true ),file(subset_1,m1_subset_1)).
thf(dt_f1_s20_fraenkel,axiom,( $true ),file(fraenkel,f1_s20_fraenkel)).
thf(dt_f2_s20_fraenkel,axiom,( $true ),file(fraenkel,f2_s20_fraenkel)).
thf(dt_f3_s20_fraenkel,axiom,( $true ),file(fraenkel,f3_s20_fraenkel)).
thf(dt_f4_s20_fraenkel,axiom,( m1_subset_1 @ f4_s20_fraenkel @ f2_s20_fraenkel ),file(fraenkel,f4_s20_fraenkel)).
thf(e1_23_1__fraenkel,axiom,( ! [A: $i] : ( ( m1_subset_1 @ A @ f1_s20_fraenkel ) => ! [B: $i] : ( ( m1_subset_1 @ B @ f2_s20_fraenkel ) => ( ( ( = @ B @ f4_s20_fraenkel ) & ( p1_s20_fraenkel @ A @ B ) ) <=> ( ( = @ B @ f4_s20_fraenkel ) & ( p1_s20_fraenkel @ A @ B ) ) ) ) ) ),file(fraenkel,e1_23_1__fraenkel)).
thf(s19_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 ) => ! [H: $i] : ( ( m1_subset_1 @ H @ E ) => ( ( C @ G @ H ) <=> ( ( = @ H @ F ) & ( B @ G @ H ) ) ) ) ) => ( = @ ( replSep2 @ ^ [G: $i] : ( m1_subset_1 @ G @ D ) @ ^ [G: $i] : ^ [H: $i] : ( m1_subset_1 @ H @ E ) @ ^ [G: $i] : ^ [H: $i] : ( A @ G @ H ) @ ^ [G: $i] : ^ [H: $i] : ( C @ G @ H ) ) @ ( replSep1 @ ^ [G: $i] : ( m1_subset_1 @ G @ D ) @ ^ [G: $i] : ( A @ G @ F ) @ ^ [G: $i] : ( B @ G @ F ) ) ) ) ) ),file(fraenkel,s19_fraenkel)).
thf(e2_23_1__fraenkel,conjecture,( = @ ( replSep2 @ ^ [A: $i] : ( m1_subset_1 @ A @ f1_s20_fraenkel ) @ ^ [A: $i] : ^ [B: $i] : ( m1_subset_1 @ B @ f2_s20_fraenkel ) @ ^ [A: $i] : ^ [B: $i] : ( f3_s20_fraenkel @ A @ B ) @ ^ [A: $i] : ^ [B: $i] : ( ( = @ B @ f4_s20_fraenkel ) & ( p1_s20_fraenkel @ A @ B ) ) ) @ ( replSep1 @ ^ [A: $i] : ( m1_subset_1 @ A @ f1_s20_fraenkel ) @ ^ [A: $i] : ( f3_s20_fraenkel @ A @ f4_s20_fraenkel ) @ ^ [A: $i] : ( p1_s20_fraenkel @ A @ f4_s20_fraenkel ) ) ),file(fraenkel,e2_23_1__fraenkel)).
thf(sethood_m1_subset_1,axiom,( ! [A: $i] : ( sethood @ ^ [X: $i] : ( m1_subset_1 @ X @ A ) ) ),file(subset_1,m1_subset_1)).
