TSTP Solution File: SET814+4 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SET814+4 : TPTP v8.1.0. Released v3.2.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n006.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue Sep 20 05:08:07 EDT 2022
% Result : Theorem 0.12s 0.39s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.11 % Problem : SET814+4 : TPTP v8.1.0. Released v3.2.0.
% 0.11/0.12 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.12/0.33 % Computer : n006.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Sat Sep 3 08:02:39 EDT 2022
% 0.12/0.33 % CPUTime :
% 0.12/0.34 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.12/0.34 Usage: tptp [options] [-file:]file
% 0.12/0.34 -h, -? prints this message.
% 0.12/0.34 -smt2 print SMT-LIB2 benchmark.
% 0.12/0.34 -m, -model generate model.
% 0.12/0.34 -p, -proof generate proof.
% 0.12/0.34 -c, -core generate unsat core of named formulas.
% 0.12/0.34 -st, -statistics display statistics.
% 0.12/0.34 -t:timeout set timeout (in second).
% 0.12/0.34 -smt2status display status in smt2 format instead of SZS.
% 0.12/0.34 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.12/0.34 -<param>:<value> configuration parameter and value.
% 0.12/0.34 -o:<output-file> file to place output in.
% 0.12/0.39 % SZS status Theorem
% 0.12/0.39 % SZS output start Proof
% 0.12/0.39 tff(member_type, type, (
% 0.12/0.39 member: ( $i * $i ) > $o)).
% 0.12/0.39 tff(tptp_fun_A_13_type, type, (
% 0.12/0.39 tptp_fun_A_13: $i)).
% 0.12/0.39 tff(tptp_fun_Y_1_type, type, (
% 0.12/0.39 tptp_fun_Y_1: ( $i * $i ) > $i)).
% 0.12/0.39 tff(tptp_fun_X_0_type, type, (
% 0.12/0.39 tptp_fun_X_0: ( $i * $i ) > $i)).
% 0.12/0.39 tff(sum_type, type, (
% 0.12/0.39 sum: $i > $i)).
% 0.12/0.39 tff(subset_type, type, (
% 0.12/0.39 subset: ( $i * $i ) > $o)).
% 0.12/0.39 tff(on_type, type, (
% 0.12/0.39 on: $i)).
% 0.12/0.39 tff(strict_well_order_type, type, (
% 0.12/0.39 strict_well_order: ( $i * $i ) > $o)).
% 0.12/0.39 tff(member_predicate_type, type, (
% 0.12/0.39 member_predicate: $i)).
% 0.12/0.39 tff(set_type, type, (
% 0.12/0.39 set: $i > $o)).
% 0.12/0.39 tff(tptp_fun_X_3_type, type, (
% 0.12/0.39 tptp_fun_X_3: $i > $i)).
% 0.12/0.39 tff(1,plain,
% 0.12/0.39 (^[A: $i, B: $i] : refl((~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))))) <=> (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(2,plain,
% 0.12/0.39 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))))) <=> ![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[1])).
% 0.12/0.39 tff(3,plain,
% 0.12/0.39 (^[A: $i, B: $i] : rewrite((~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))))) <=> (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(4,plain,
% 0.12/0.39 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))))) <=> ![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[3])).
% 0.12/0.39 tff(5,plain,
% 0.12/0.39 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))))) <=> ![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))),
% 0.12/0.39 inference(transitivity,[status(thm)],[4, 2])).
% 0.12/0.39 tff(6,plain,
% 0.12/0.39 (^[A: $i, B: $i] : trans(monotonicity(rewrite(((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B))) <=> ((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))), rewrite((subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))) <=> (subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))), ((((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B))) & (subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))) <=> (((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B))) & (subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))))), rewrite((((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B))) & (subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))) <=> (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))), ((((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B))) & (subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))) <=> (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(7,plain,
% 0.12/0.40 (![A: $i, B: $i] : (((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B))) & (subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))) <=> ![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[6])).
% 0.12/0.40 tff(8,plain,
% 0.12/0.40 (![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B))) <=> ![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B)))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(9,plain,
% 0.12/0.40 (^[A: $i, B: $i] : rewrite((subset(A, B) <=> ![X: $i] : (member(X, A) => member(X, B))) <=> (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B))))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(10,plain,
% 0.12/0.40 (![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : (member(X, A) => member(X, B))) <=> ![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B)))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[9])).
% 0.12/0.40 tff(11,axiom,(![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : (member(X, A) => member(X, B)))), file('/export/starexec/sandbox/benchmark/Axioms/SET006+0.ax','subset')).
% 0.12/0.40 tff(12,plain,
% 0.12/0.40 (![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B)))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[11, 10])).
% 0.12/0.40 tff(13,plain,
% 0.12/0.40 (![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B)))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[12, 8])).
% 0.12/0.40 tff(14,plain,(
% 0.12/0.40 ![A: $i, B: $i] : (((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B))) & (subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))),
% 0.12/0.40 inference(skolemize,[status(sab)],[13])).
% 0.12/0.40 tff(15,plain,
% 0.12/0.40 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[14, 7])).
% 0.12/0.40 tff(16,plain,
% 0.12/0.40 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[15, 5])).
% 0.12/0.40 tff(17,plain,
% 0.12/0.40 ((~![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))) | (~((~((~subset(tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))), A!13)) | ![X: $i] : ((~member(X, tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))))) | member(X, A!13)))) | (~(subset(tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))), A!13) | (~((~member(tptp_fun_X_0(A!13, tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13)))), tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))))) | member(tptp_fun_X_0(A!13, tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13)))), A!13)))))))),
% 0.12/0.40 inference(quant_inst,[status(thm)],[])).
% 0.12/0.40 tff(18,plain,
% 0.12/0.40 (~((~((~subset(tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))), A!13)) | ![X: $i] : ((~member(X, tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))))) | member(X, A!13)))) | (~(subset(tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))), A!13) | (~((~member(tptp_fun_X_0(A!13, tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13)))), tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))))) | member(tptp_fun_X_0(A!13, tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13)))), A!13))))))),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[17, 16])).
% 0.12/0.40 tff(19,plain,
% 0.12/0.40 (((~((~subset(tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))), A!13)) | ![X: $i] : ((~member(X, tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))))) | member(X, A!13)))) | (~(subset(tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))), A!13) | (~((~member(tptp_fun_X_0(A!13, tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13)))), tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))))) | member(tptp_fun_X_0(A!13, tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13)))), A!13)))))) | ((~subset(tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))), A!13)) | ![X: $i] : ((~member(X, tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))))) | member(X, A!13)))),
% 0.12/0.40 inference(tautology,[status(thm)],[])).
% 0.12/0.40 tff(20,plain,
% 0.12/0.40 ((~subset(tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))), A!13)) | ![X: $i] : ((~member(X, tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))))) | member(X, A!13))),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[19, 18])).
% 0.12/0.40 tff(21,plain,
% 0.12/0.40 (^[X: $i, A: $i] : refl((~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y))))))) <=> (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y))))))))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(22,plain,
% 0.12/0.40 (![X: $i, A: $i] : (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y))))))) <=> ![X: $i, A: $i] : (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y)))))))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[21])).
% 0.12/0.40 tff(23,plain,
% 0.12/0.40 (^[X: $i, A: $i] : rewrite((~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y))))))) <=> (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y))))))))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(24,plain,
% 0.12/0.40 (![X: $i, A: $i] : (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y))))))) <=> ![X: $i, A: $i] : (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y)))))))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[23])).
% 0.12/0.40 tff(25,plain,
% 0.12/0.40 (![X: $i, A: $i] : (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y))))))) <=> ![X: $i, A: $i] : (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y)))))))),
% 0.12/0.40 inference(transitivity,[status(thm)],[24, 22])).
% 0.12/0.40 tff(26,plain,
% 0.12/0.40 (^[X: $i, A: $i] : trans(monotonicity(rewrite(((~member(X, sum(A))) | (member(tptp_fun_Y_1(A, X), A) & member(X, tptp_fun_Y_1(A, X)))) <=> ((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))), rewrite((member(X, sum(A)) | ![Y: $i] : (~(member(Y, A) & member(X, Y)))) <=> (member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y))))), ((((~member(X, sum(A))) | (member(tptp_fun_Y_1(A, X), A) & member(X, tptp_fun_Y_1(A, X)))) & (member(X, sum(A)) | ![Y: $i] : (~(member(Y, A) & member(X, Y))))) <=> (((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X)))))) & (member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y))))))), rewrite((((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X)))))) & (member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y))))) <=> (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y)))))))), ((((~member(X, sum(A))) | (member(tptp_fun_Y_1(A, X), A) & member(X, tptp_fun_Y_1(A, X)))) & (member(X, sum(A)) | ![Y: $i] : (~(member(Y, A) & member(X, Y))))) <=> (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y)))))))))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(27,plain,
% 0.12/0.40 (![X: $i, A: $i] : (((~member(X, sum(A))) | (member(tptp_fun_Y_1(A, X), A) & member(X, tptp_fun_Y_1(A, X)))) & (member(X, sum(A)) | ![Y: $i] : (~(member(Y, A) & member(X, Y))))) <=> ![X: $i, A: $i] : (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y)))))))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[26])).
% 0.12/0.40 tff(28,plain,
% 0.12/0.40 (![X: $i, A: $i] : (member(X, sum(A)) <=> ?[Y: $i] : (member(Y, A) & member(X, Y))) <=> ![X: $i, A: $i] : (member(X, sum(A)) <=> ?[Y: $i] : (member(Y, A) & member(X, Y)))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(29,axiom,(![X: $i, A: $i] : (member(X, sum(A)) <=> ?[Y: $i] : (member(Y, A) & member(X, Y)))), file('/export/starexec/sandbox/benchmark/Axioms/SET006+0.ax','sum')).
% 0.12/0.40 tff(30,plain,
% 0.12/0.40 (![X: $i, A: $i] : (member(X, sum(A)) <=> ?[Y: $i] : (member(Y, A) & member(X, Y)))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[29, 28])).
% 0.12/0.40 tff(31,plain,(
% 0.12/0.40 ![X: $i, A: $i] : (((~member(X, sum(A))) | (member(tptp_fun_Y_1(A, X), A) & member(X, tptp_fun_Y_1(A, X)))) & (member(X, sum(A)) | ![Y: $i] : (~(member(Y, A) & member(X, Y)))))),
% 0.12/0.40 inference(skolemize,[status(sab)],[30])).
% 0.12/0.40 tff(32,plain,
% 0.12/0.40 (![X: $i, A: $i] : (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y)))))))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[31, 27])).
% 0.12/0.40 tff(33,plain,
% 0.12/0.40 (![X: $i, A: $i] : (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y)))))))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[32, 25])).
% 0.12/0.40 tff(34,plain,
% 0.12/0.40 ((~![X: $i, A: $i] : (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y)))))))) | (~((~((~member(tptp_fun_X_0(A!13, sum(A!13)), sum(A!13))) | (~((~member(tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))), A!13)) | (~member(tptp_fun_X_0(A!13, sum(A!13)), tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))))))))) | (~(member(tptp_fun_X_0(A!13, sum(A!13)), sum(A!13)) | ![Y: $i] : ((~member(Y, A!13)) | (~member(tptp_fun_X_0(A!13, sum(A!13)), Y)))))))),
% 0.12/0.40 inference(quant_inst,[status(thm)],[])).
% 0.12/0.40 tff(35,plain,
% 0.12/0.40 (~((~((~member(tptp_fun_X_0(A!13, sum(A!13)), sum(A!13))) | (~((~member(tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))), A!13)) | (~member(tptp_fun_X_0(A!13, sum(A!13)), tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))))))))) | (~(member(tptp_fun_X_0(A!13, sum(A!13)), sum(A!13)) | ![Y: $i] : ((~member(Y, A!13)) | (~member(tptp_fun_X_0(A!13, sum(A!13)), Y))))))),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[34, 33])).
% 0.12/0.40 tff(36,plain,
% 0.12/0.40 (((~((~member(tptp_fun_X_0(A!13, sum(A!13)), sum(A!13))) | (~((~member(tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))), A!13)) | (~member(tptp_fun_X_0(A!13, sum(A!13)), tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))))))))) | (~(member(tptp_fun_X_0(A!13, sum(A!13)), sum(A!13)) | ![Y: $i] : ((~member(Y, A!13)) | (~member(tptp_fun_X_0(A!13, sum(A!13)), Y)))))) | ((~member(tptp_fun_X_0(A!13, sum(A!13)), sum(A!13))) | (~((~member(tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))), A!13)) | (~member(tptp_fun_X_0(A!13, sum(A!13)), tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))))))))),
% 0.12/0.40 inference(tautology,[status(thm)],[])).
% 0.12/0.40 tff(37,plain,
% 0.12/0.40 ((~member(tptp_fun_X_0(A!13, sum(A!13)), sum(A!13))) | (~((~member(tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))), A!13)) | (~member(tptp_fun_X_0(A!13, sum(A!13)), tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13)))))))),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[36, 35])).
% 0.12/0.40 tff(38,plain,
% 0.12/0.40 ((~![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))) | (~((~((~subset(sum(A!13), A!13)) | ![X: $i] : ((~member(X, sum(A!13))) | member(X, A!13)))) | (~(subset(sum(A!13), A!13) | (~((~member(tptp_fun_X_0(A!13, sum(A!13)), sum(A!13))) | member(tptp_fun_X_0(A!13, sum(A!13)), A!13)))))))),
% 0.12/0.40 inference(quant_inst,[status(thm)],[])).
% 0.12/0.40 tff(39,plain,
% 0.12/0.40 (~((~((~subset(sum(A!13), A!13)) | ![X: $i] : ((~member(X, sum(A!13))) | member(X, A!13)))) | (~(subset(sum(A!13), A!13) | (~((~member(tptp_fun_X_0(A!13, sum(A!13)), sum(A!13))) | member(tptp_fun_X_0(A!13, sum(A!13)), A!13))))))),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[38, 16])).
% 0.12/0.40 tff(40,plain,
% 0.12/0.40 (((~((~subset(sum(A!13), A!13)) | ![X: $i] : ((~member(X, sum(A!13))) | member(X, A!13)))) | (~(subset(sum(A!13), A!13) | (~((~member(tptp_fun_X_0(A!13, sum(A!13)), sum(A!13))) | member(tptp_fun_X_0(A!13, sum(A!13)), A!13)))))) | (subset(sum(A!13), A!13) | (~((~member(tptp_fun_X_0(A!13, sum(A!13)), sum(A!13))) | member(tptp_fun_X_0(A!13, sum(A!13)), A!13))))),
% 0.12/0.40 inference(tautology,[status(thm)],[])).
% 0.12/0.40 tff(41,plain,
% 0.12/0.40 (subset(sum(A!13), A!13) | (~((~member(tptp_fun_X_0(A!13, sum(A!13)), sum(A!13))) | member(tptp_fun_X_0(A!13, sum(A!13)), A!13)))),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[40, 39])).
% 0.12/0.40 tff(42,plain,
% 0.12/0.40 ((~![A: $i] : ((~member(A, on)) | subset(sum(A), A))) <=> (~![A: $i] : ((~member(A, on)) | subset(sum(A), A)))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(43,plain,
% 0.12/0.40 ((~![A: $i] : (member(A, on) => subset(sum(A), A))) <=> (~![A: $i] : ((~member(A, on)) | subset(sum(A), A)))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(44,axiom,(~![A: $i] : (member(A, on) => subset(sum(A), A))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','thV14')).
% 0.12/0.40 tff(45,plain,
% 0.12/0.40 (~![A: $i] : ((~member(A, on)) | subset(sum(A), A))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[44, 43])).
% 0.12/0.40 tff(46,plain,
% 0.12/0.40 (~![A: $i] : ((~member(A, on)) | subset(sum(A), A))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[45, 42])).
% 0.12/0.40 tff(47,plain,
% 0.12/0.40 (~![A: $i] : ((~member(A, on)) | subset(sum(A), A))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[46, 42])).
% 0.12/0.40 tff(48,plain,
% 0.12/0.40 (~![A: $i] : ((~member(A, on)) | subset(sum(A), A))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[47, 42])).
% 0.12/0.40 tff(49,plain,
% 0.12/0.40 (~![A: $i] : ((~member(A, on)) | subset(sum(A), A))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[48, 42])).
% 0.12/0.40 tff(50,plain,
% 0.12/0.40 (~![A: $i] : ((~member(A, on)) | subset(sum(A), A))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[49, 42])).
% 0.12/0.40 tff(51,plain,
% 0.12/0.40 (~![A: $i] : ((~member(A, on)) | subset(sum(A), A))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[50, 42])).
% 0.12/0.40 tff(52,plain,(
% 0.12/0.40 ~((~member(A!13, on)) | subset(sum(A!13), A!13))),
% 0.12/0.40 inference(skolemize,[status(sab)],[51])).
% 0.12/0.40 tff(53,plain,
% 0.12/0.40 (~subset(sum(A!13), A!13)),
% 0.12/0.40 inference(or_elim,[status(thm)],[52])).
% 0.12/0.40 tff(54,plain,
% 0.12/0.40 ((~(subset(sum(A!13), A!13) | (~((~member(tptp_fun_X_0(A!13, sum(A!13)), sum(A!13))) | member(tptp_fun_X_0(A!13, sum(A!13)), A!13))))) | subset(sum(A!13), A!13) | (~((~member(tptp_fun_X_0(A!13, sum(A!13)), sum(A!13))) | member(tptp_fun_X_0(A!13, sum(A!13)), A!13)))),
% 0.12/0.40 inference(tautology,[status(thm)],[])).
% 0.12/0.40 tff(55,plain,
% 0.12/0.40 ((~(subset(sum(A!13), A!13) | (~((~member(tptp_fun_X_0(A!13, sum(A!13)), sum(A!13))) | member(tptp_fun_X_0(A!13, sum(A!13)), A!13))))) | (~((~member(tptp_fun_X_0(A!13, sum(A!13)), sum(A!13))) | member(tptp_fun_X_0(A!13, sum(A!13)), A!13)))),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[54, 53])).
% 0.12/0.40 tff(56,plain,
% 0.12/0.40 (~((~member(tptp_fun_X_0(A!13, sum(A!13)), sum(A!13))) | member(tptp_fun_X_0(A!13, sum(A!13)), A!13))),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[55, 41])).
% 0.12/0.40 tff(57,plain,
% 0.12/0.40 (((~member(tptp_fun_X_0(A!13, sum(A!13)), sum(A!13))) | member(tptp_fun_X_0(A!13, sum(A!13)), A!13)) | member(tptp_fun_X_0(A!13, sum(A!13)), sum(A!13))),
% 0.12/0.40 inference(tautology,[status(thm)],[])).
% 0.12/0.40 tff(58,plain,
% 0.12/0.40 (member(tptp_fun_X_0(A!13, sum(A!13)), sum(A!13))),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[57, 56])).
% 0.19/0.41 tff(59,plain,
% 0.19/0.41 ((~((~member(tptp_fun_X_0(A!13, sum(A!13)), sum(A!13))) | (~((~member(tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))), A!13)) | (~member(tptp_fun_X_0(A!13, sum(A!13)), tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))))))))) | (~member(tptp_fun_X_0(A!13, sum(A!13)), sum(A!13))) | (~((~member(tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))), A!13)) | (~member(tptp_fun_X_0(A!13, sum(A!13)), tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13)))))))),
% 0.19/0.41 inference(tautology,[status(thm)],[])).
% 0.19/0.41 tff(60,plain,
% 0.19/0.41 ((~((~member(tptp_fun_X_0(A!13, sum(A!13)), sum(A!13))) | (~((~member(tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))), A!13)) | (~member(tptp_fun_X_0(A!13, sum(A!13)), tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))))))))) | (~((~member(tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))), A!13)) | (~member(tptp_fun_X_0(A!13, sum(A!13)), tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13)))))))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[59, 58])).
% 0.19/0.41 tff(61,plain,
% 0.19/0.41 (~((~member(tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))), A!13)) | (~member(tptp_fun_X_0(A!13, sum(A!13)), tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))))))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[60, 37])).
% 0.19/0.41 tff(62,plain,
% 0.19/0.41 (((~member(tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))), A!13)) | (~member(tptp_fun_X_0(A!13, sum(A!13)), tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13)))))) | member(tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))), A!13)),
% 0.19/0.41 inference(tautology,[status(thm)],[])).
% 0.19/0.41 tff(63,plain,
% 0.19/0.41 (member(tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))), A!13)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[62, 61])).
% 0.19/0.41 tff(64,plain,
% 0.19/0.41 (^[A: $i] : rewrite((~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))))) <=> (~((~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))) | (~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(65,plain,
% 0.19/0.41 (![A: $i] : (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))))) <=> ![A: $i] : (~((~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))) | (~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A)))))))))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[64])).
% 0.19/0.41 tff(66,plain,
% 0.19/0.41 (^[A: $i] : refl((~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))))) <=> (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(67,plain,
% 0.19/0.41 (![A: $i] : (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))))) <=> ![A: $i] : (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))))))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[66])).
% 0.19/0.41 tff(68,plain,
% 0.19/0.41 (^[A: $i] : rewrite((~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))))) <=> (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(69,plain,
% 0.19/0.41 (![A: $i] : (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))))) <=> ![A: $i] : (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))))))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[68])).
% 0.19/0.41 tff(70,plain,
% 0.19/0.41 (![A: $i] : (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))))) <=> ![A: $i] : (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))))))),
% 0.19/0.41 inference(transitivity,[status(thm)],[69, 67])).
% 0.19/0.41 tff(71,plain,
% 0.19/0.41 (^[A: $i] : trans(monotonicity(rewrite(((~member(A, on)) | (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A)))) <=> ((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))), rewrite((member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))) <=> (member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))), ((((~member(A, on)) | (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A)))) & (member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))) <=> (((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A)))))) & (member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))))), rewrite((((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A)))))) & (member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))) <=> (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))))))), ((((~member(A, on)) | (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A)))) & (member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))) <=> (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))))))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(72,plain,
% 0.19/0.41 (![A: $i] : (((~member(A, on)) | (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A)))) & (member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))) <=> ![A: $i] : (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))))))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[71])).
% 0.19/0.41 tff(73,plain,
% 0.19/0.41 (^[A: $i] : rewrite((((~member(A, on)) | (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A)))) & (member(A, on) | ((~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))))) <=> (((~member(A, on)) | (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A)))) & (member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(74,plain,
% 0.19/0.41 (![A: $i] : (((~member(A, on)) | (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A)))) & (member(A, on) | ((~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))))) <=> ![A: $i] : (((~member(A, on)) | (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A)))) & (member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[73])).
% 0.19/0.41 tff(75,plain,
% 0.19/0.41 (![A: $i] : (member(A, on) <=> (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A)))) <=> ![A: $i] : (member(A, on) <=> (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A))))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(76,plain,
% 0.19/0.41 (^[A: $i] : rewrite((member(A, on) <=> ((set(A) & strict_well_order(member_predicate, A)) & ![X: $i] : (member(X, A) => subset(X, A)))) <=> (member(A, on) <=> (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A)))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(77,plain,
% 0.19/0.41 (![A: $i] : (member(A, on) <=> ((set(A) & strict_well_order(member_predicate, A)) & ![X: $i] : (member(X, A) => subset(X, A)))) <=> ![A: $i] : (member(A, on) <=> (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A))))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[76])).
% 0.19/0.41 tff(78,axiom,(![A: $i] : (member(A, on) <=> ((set(A) & strict_well_order(member_predicate, A)) & ![X: $i] : (member(X, A) => subset(X, A))))), file('/export/starexec/sandbox/benchmark/Axioms/SET006+4.ax','ordinal_number')).
% 0.19/0.41 tff(79,plain,
% 0.19/0.41 (![A: $i] : (member(A, on) <=> (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[78, 77])).
% 0.19/0.41 tff(80,plain,
% 0.19/0.41 (![A: $i] : (member(A, on) <=> (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[79, 75])).
% 0.19/0.41 tff(81,plain,(
% 0.19/0.41 ![A: $i] : (((~member(A, on)) | (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A)))) & (member(A, on) | ((~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))))),
% 0.19/0.41 inference(skolemize,[status(sab)],[80])).
% 0.19/0.41 tff(82,plain,
% 0.19/0.41 (![A: $i] : (((~member(A, on)) | (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A)))) & (member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[81, 74])).
% 0.19/0.41 tff(83,plain,
% 0.19/0.41 (![A: $i] : (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[82, 72])).
% 0.19/0.41 tff(84,plain,
% 0.19/0.41 (![A: $i] : (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[83, 70])).
% 0.19/0.41 tff(85,plain,
% 0.19/0.41 (![A: $i] : (~((~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))) | (~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A)))))))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[84, 65])).
% 0.19/0.41 tff(86,plain,
% 0.19/0.41 ((~![A: $i] : (~((~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))) | (~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A)))))))))) | (~((~(member(A!13, on) | (~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~((~member(tptp_fun_X_3(A!13), A!13)) | subset(tptp_fun_X_3(A!13), A!13))))) | (~((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13)))))))))),
% 0.19/0.41 inference(quant_inst,[status(thm)],[])).
% 0.19/0.41 tff(87,plain,
% 0.19/0.41 (~((~(member(A!13, on) | (~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~((~member(tptp_fun_X_3(A!13), A!13)) | subset(tptp_fun_X_3(A!13), A!13))))) | (~((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13))))))))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[86, 85])).
% 0.19/0.41 tff(88,plain,
% 0.19/0.41 (((~(member(A!13, on) | (~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~((~member(tptp_fun_X_3(A!13), A!13)) | subset(tptp_fun_X_3(A!13), A!13))))) | (~((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13)))))))) | ((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13))))))),
% 0.19/0.41 inference(tautology,[status(thm)],[])).
% 0.19/0.41 tff(89,plain,
% 0.19/0.41 ((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13)))))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[88, 87])).
% 0.19/0.41 tff(90,plain,
% 0.19/0.41 (member(A!13, on)),
% 0.19/0.41 inference(or_elim,[status(thm)],[52])).
% 0.19/0.41 tff(91,plain,
% 0.19/0.41 ((~((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13))))))) | (~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13)))))),
% 0.19/0.41 inference(tautology,[status(thm)],[])).
% 0.19/0.41 tff(92,plain,
% 0.19/0.41 ((~((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13))))))) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13)))))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[91, 90])).
% 0.19/0.41 tff(93,plain,
% 0.19/0.41 (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13))))),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[92, 89])).
% 0.19/0.42 tff(94,plain,
% 0.19/0.42 (((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13)))) | ![X: $i] : ((~member(X, A!13)) | subset(X, A!13))),
% 0.19/0.42 inference(tautology,[status(thm)],[])).
% 0.19/0.42 tff(95,plain,
% 0.19/0.42 (![X: $i] : ((~member(X, A!13)) | subset(X, A!13))),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[94, 93])).
% 0.19/0.42 tff(96,plain,
% 0.19/0.42 (((~![X: $i] : ((~member(X, A!13)) | subset(X, A!13))) | ((~member(tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))), A!13)) | subset(tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))), A!13))) <=> ((~![X: $i] : ((~member(X, A!13)) | subset(X, A!13))) | (~member(tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))), A!13)) | subset(tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))), A!13))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(97,plain,
% 0.19/0.42 ((~![X: $i] : ((~member(X, A!13)) | subset(X, A!13))) | ((~member(tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))), A!13)) | subset(tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))), A!13))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(98,plain,
% 0.19/0.42 ((~![X: $i] : ((~member(X, A!13)) | subset(X, A!13))) | (~member(tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))), A!13)) | subset(tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))), A!13)),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[97, 96])).
% 0.19/0.42 tff(99,plain,
% 0.19/0.42 (subset(tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))), A!13)),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[98, 95, 63])).
% 0.19/0.42 tff(100,plain,
% 0.19/0.42 ((~((~subset(tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))), A!13)) | ![X: $i] : ((~member(X, tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))))) | member(X, A!13)))) | (~subset(tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))), A!13)) | ![X: $i] : ((~member(X, tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))))) | member(X, A!13))),
% 0.19/0.42 inference(tautology,[status(thm)],[])).
% 0.19/0.42 tff(101,plain,
% 0.19/0.42 ((~((~subset(tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))), A!13)) | ![X: $i] : ((~member(X, tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))))) | member(X, A!13)))) | ![X: $i] : ((~member(X, tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))))) | member(X, A!13))),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[100, 99])).
% 0.19/0.42 tff(102,plain,
% 0.19/0.42 (![X: $i] : ((~member(X, tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))))) | member(X, A!13))),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[101, 20])).
% 0.19/0.42 tff(103,plain,
% 0.19/0.42 (((~member(tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))), A!13)) | (~member(tptp_fun_X_0(A!13, sum(A!13)), tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13)))))) | member(tptp_fun_X_0(A!13, sum(A!13)), tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))))),
% 0.19/0.42 inference(tautology,[status(thm)],[])).
% 0.19/0.42 tff(104,plain,
% 0.19/0.42 (member(tptp_fun_X_0(A!13, sum(A!13)), tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))))),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[103, 61])).
% 0.19/0.42 tff(105,plain,
% 0.19/0.42 (((~member(tptp_fun_X_0(A!13, sum(A!13)), sum(A!13))) | member(tptp_fun_X_0(A!13, sum(A!13)), A!13)) | (~member(tptp_fun_X_0(A!13, sum(A!13)), A!13))),
% 0.19/0.42 inference(tautology,[status(thm)],[])).
% 0.19/0.42 tff(106,plain,
% 0.19/0.42 (~member(tptp_fun_X_0(A!13, sum(A!13)), A!13)),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[105, 56])).
% 0.19/0.42 tff(107,plain,
% 0.19/0.42 (((~![X: $i] : ((~member(X, tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))))) | member(X, A!13))) | ((~member(tptp_fun_X_0(A!13, sum(A!13)), tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))))) | member(tptp_fun_X_0(A!13, sum(A!13)), A!13))) <=> ((~![X: $i] : ((~member(X, tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))))) | member(X, A!13))) | (~member(tptp_fun_X_0(A!13, sum(A!13)), tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))))) | member(tptp_fun_X_0(A!13, sum(A!13)), A!13))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(108,plain,
% 0.19/0.42 ((~![X: $i] : ((~member(X, tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))))) | member(X, A!13))) | ((~member(tptp_fun_X_0(A!13, sum(A!13)), tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))))) | member(tptp_fun_X_0(A!13, sum(A!13)), A!13))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(109,plain,
% 0.19/0.42 ((~![X: $i] : ((~member(X, tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))))) | member(X, A!13))) | (~member(tptp_fun_X_0(A!13, sum(A!13)), tptp_fun_Y_1(A!13, tptp_fun_X_0(A!13, sum(A!13))))) | member(tptp_fun_X_0(A!13, sum(A!13)), A!13)),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[108, 107])).
% 0.19/0.42 tff(110,plain,
% 0.19/0.42 ($false),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[109, 106, 104, 102])).
% 0.19/0.42 % SZS output end Proof
%------------------------------------------------------------------------------