%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SEU147+3 : TPTP v9.0.0. Bugfixed v4.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n004.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 : Sat Jun 21 05:27:01 AM UTC 2025
% Result : Theorem 0.20s 0.45s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : SEU147+3 : TPTP v9.0.0. Bugfixed v4.0.0.
% 0.03/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n004.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 : Fri Jun 20 06:07:23 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.20/0.45 % SZS status Theorem
% 0.20/0.45 % SZS output start Proof
% 0.20/0.45 tff(subset_type, type, (
% 0.20/0.45 subset: ( $i * $i ) > $o)).
% 0.20/0.45 tff(empty_set_type, type, (
% 0.20/0.45 empty_set: $i)).
% 0.20/0.45 tff(tptp_fun_C_1_type, type, (
% 0.20/0.45 tptp_fun_C_1: ( $i * $i ) > $i)).
% 0.20/0.45 tff(singleton_type, type, (
% 0.20/0.45 singleton: $i > $i)).
% 0.20/0.45 tff(in_type, type, (
% 0.20/0.45 in: ( $i * $i ) > $o)).
% 0.20/0.45 tff(tptp_fun_C_0_type, type, (
% 0.20/0.45 tptp_fun_C_0: ( $i * $i ) > $i)).
% 0.20/0.45 tff(powerset_type, type, (
% 0.20/0.45 powerset: $i > $i)).
% 0.20/0.45 tff(1,assumption,(~(tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set)), introduced(assumption)).
% 0.20/0.45 tff(2,assumption,(subset(tptp_fun_C_1(singleton(empty_set), empty_set), empty_set)), introduced(assumption)).
% 0.20/0.45 tff(3,plain,
% 0.20/0.45 (^[A: $i] : refl(((~subset(A, empty_set)) | (A = empty_set)) <=> ((~subset(A, empty_set)) | (A = empty_set)))),
% 0.20/0.45 inference(bind,[status(th)],[])).
% 0.20/0.45 tff(4,plain,
% 0.20/0.45 (![A: $i] : ((~subset(A, empty_set)) | (A = empty_set)) <=> ![A: $i] : ((~subset(A, empty_set)) | (A = empty_set))),
% 0.20/0.45 inference(quant_intro,[status(thm)],[3])).
% 0.20/0.45 tff(5,plain,
% 0.20/0.45 (![A: $i] : ((~subset(A, empty_set)) | (A = empty_set)) <=> ![A: $i] : ((~subset(A, empty_set)) | (A = empty_set))),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(6,plain,
% 0.20/0.45 (^[A: $i] : rewrite((subset(A, empty_set) => (A = empty_set)) <=> ((~subset(A, empty_set)) | (A = empty_set)))),
% 0.20/0.45 inference(bind,[status(th)],[])).
% 0.20/0.45 tff(7,plain,
% 0.20/0.45 (![A: $i] : (subset(A, empty_set) => (A = empty_set)) <=> ![A: $i] : ((~subset(A, empty_set)) | (A = empty_set))),
% 0.20/0.45 inference(quant_intro,[status(thm)],[6])).
% 0.20/0.45 tff(8,axiom,(![A: $i] : (subset(A, empty_set) => (A = empty_set))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t3_xboole_1')).
% 0.20/0.45 tff(9,plain,
% 0.20/0.45 (![A: $i] : ((~subset(A, empty_set)) | (A = empty_set))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[8, 7])).
% 0.20/0.45 tff(10,plain,
% 0.20/0.45 (![A: $i] : ((~subset(A, empty_set)) | (A = empty_set))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[9, 5])).
% 0.20/0.45 tff(11,plain,(
% 0.20/0.45 ![A: $i] : ((~subset(A, empty_set)) | (A = empty_set))),
% 0.20/0.45 inference(skolemize,[status(sab)],[10])).
% 0.20/0.45 tff(12,plain,
% 0.20/0.45 (![A: $i] : ((~subset(A, empty_set)) | (A = empty_set))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[11, 4])).
% 0.20/0.45 tff(13,plain,
% 0.20/0.45 (((~![A: $i] : ((~subset(A, empty_set)) | (A = empty_set))) | ((~subset(tptp_fun_C_1(singleton(empty_set), empty_set), empty_set)) | (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set))) <=> ((~![A: $i] : ((~subset(A, empty_set)) | (A = empty_set))) | (~subset(tptp_fun_C_1(singleton(empty_set), empty_set), empty_set)) | (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set))),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(14,plain,
% 0.20/0.45 ((~![A: $i] : ((~subset(A, empty_set)) | (A = empty_set))) | ((~subset(tptp_fun_C_1(singleton(empty_set), empty_set), empty_set)) | (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set))),
% 0.20/0.45 inference(quant_inst,[status(thm)],[])).
% 0.20/0.45 tff(15,plain,
% 0.20/0.45 ((~![A: $i] : ((~subset(A, empty_set)) | (A = empty_set))) | (~subset(tptp_fun_C_1(singleton(empty_set), empty_set), empty_set)) | (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set)),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[14, 13])).
% 0.20/0.45 tff(16,plain,
% 0.20/0.45 ($false),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[15, 12, 2, 1])).
% 0.20/0.45 tff(17,plain,((~subset(tptp_fun_C_1(singleton(empty_set), empty_set), empty_set)) | (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set)), inference(lemma,lemma(discharge,[]))).
% 0.20/0.45 tff(18,plain,
% 0.20/0.45 (~subset(tptp_fun_C_1(singleton(empty_set), empty_set), empty_set)),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[17, 1])).
% 0.20/0.45 tff(19,plain,
% 0.20/0.45 (^[A: $i, B: $i, C: $i] : refl((~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))))),
% 0.20/0.45 inference(bind,[status(th)],[])).
% 0.20/0.45 tff(20,plain,
% 0.20/0.45 (![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.20/0.45 inference(quant_intro,[status(thm)],[19])).
% 0.20/0.45 tff(21,plain,
% 0.20/0.45 (![A: $i, B: $i] : ![C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.20/0.45 inference(pull_quant,[status(thm)],[])).
% 0.20/0.45 tff(22,plain,
% 0.20/0.45 (^[A: $i, B: $i] : trans(monotonicity(trans(monotonicity(trans(monotonicity(pull_quant(((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) <=> ![C: $i] : ((~(B = singleton(A))) | (in(C, B) <=> (C = A)))), ((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) <=> (~![C: $i] : ((~(B = singleton(A))) | (in(C, B) <=> (C = A)))))), pull_quant((~![C: $i] : ((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) <=> ?[C: $i] : (~((~(B = singleton(A))) | (in(C, B) <=> (C = A))))), ((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) <=> ?[C: $i] : (~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))))), (((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))) <=> (?[C: $i] : (~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))), pull_quant((?[C: $i] : (~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))) <=> ?[C: $i] : ((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))), (((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))) <=> ?[C: $i] : ((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))), ((~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> (~?[C: $i] : ((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))))), pull_quant((~?[C: $i] : ((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))), ((~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))))),
% 0.20/0.45 inference(bind,[status(th)],[])).
% 0.20/0.45 tff(23,plain,
% 0.20/0.45 (![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![A: $i, B: $i] : ![C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.20/0.45 inference(quant_intro,[status(thm)],[22])).
% 0.20/0.45 tff(24,plain,
% 0.20/0.45 (![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.20/0.45 inference(transitivity,[status(thm)],[23, 21])).
% 0.20/0.45 tff(25,plain,
% 0.20/0.45 (![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.20/0.45 inference(transitivity,[status(thm)],[24, 20])).
% 0.20/0.45 tff(26,plain,
% 0.20/0.45 (^[A: $i, B: $i] : rewrite((~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))))),
% 0.20/0.45 inference(bind,[status(th)],[])).
% 0.20/0.45 tff(27,plain,
% 0.20/0.45 (![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.20/0.45 inference(quant_intro,[status(thm)],[26])).
% 0.20/0.45 tff(28,plain,
% 0.20/0.45 (![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.20/0.45 inference(transitivity,[status(thm)],[27, 25])).
% 0.20/0.45 tff(29,plain,
% 0.20/0.45 (^[A: $i, B: $i] : trans(monotonicity(rewrite(((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) <=> ((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))), trans(monotonicity(rewrite((~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))) <=> (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))), (((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))) <=> ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))), rewrite(((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))) <=> ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))), (((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))) <=> ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))), ((((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) & ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))) <=> (((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) & ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))), rewrite((((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) & ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))) <=> (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))), ((((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) & ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))) <=> (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))))),
% 0.20/0.45 inference(bind,[status(th)],[])).
% 0.20/0.45 tff(30,plain,
% 0.20/0.45 (![A: $i, B: $i] : (((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) & ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))) <=> ![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.20/0.45 inference(quant_intro,[status(thm)],[29])).
% 0.20/0.45 tff(31,plain,
% 0.20/0.45 (![A: $i, B: $i] : ((B = singleton(A)) <=> ![C: $i] : (in(C, B) <=> (C = A))) <=> ![A: $i, B: $i] : ((B = singleton(A)) <=> ![C: $i] : (in(C, B) <=> (C = A)))),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(32,axiom,(![A: $i, B: $i] : ((B = singleton(A)) <=> ![C: $i] : (in(C, B) <=> (C = A)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','d1_tarski')).
% 0.20/0.45 tff(33,plain,
% 0.20/0.45 (![A: $i, B: $i] : ((B = singleton(A)) <=> ![C: $i] : (in(C, B) <=> (C = A)))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[32, 31])).
% 0.20/0.45 tff(34,plain,(
% 0.20/0.45 ![A: $i, B: $i] : (((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) & ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))),
% 0.20/0.45 inference(skolemize,[status(sab)],[33])).
% 0.20/0.45 tff(35,plain,
% 0.20/0.45 (![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[34, 30])).
% 0.20/0.45 tff(36,plain,
% 0.20/0.45 (![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[35, 28])).
% 0.20/0.45 tff(37,plain,
% 0.20/0.45 (((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set))) <=> ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set)))),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(38,plain,
% 0.20/0.45 ((~(~(in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set)))) <=> (in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set))),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(39,plain,
% 0.20/0.45 (((~(in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set))) | $false) <=> (~(in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set)))),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(40,plain,
% 0.20/0.45 ((~$true) <=> $false),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(41,plain,
% 0.20/0.45 (($true | (~(in(tptp_fun_C_0(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_0(singleton(empty_set), empty_set) = empty_set)))) <=> $true),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(42,plain,
% 0.20/0.45 ((~(in(tptp_fun_C_0(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_0(singleton(empty_set), empty_set) = empty_set))) <=> (~(in(tptp_fun_C_0(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_0(singleton(empty_set), empty_set) = empty_set)))),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(43,plain,
% 0.20/0.45 ((singleton(empty_set) = singleton(empty_set)) <=> $true),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(44,plain,
% 0.20/0.45 (((singleton(empty_set) = singleton(empty_set)) | (~(in(tptp_fun_C_0(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_0(singleton(empty_set), empty_set) = empty_set)))) <=> ($true | (~(in(tptp_fun_C_0(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_0(singleton(empty_set), empty_set) = empty_set))))),
% 0.20/0.45 inference(monotonicity,[status(thm)],[43, 42])).
% 0.20/0.45 tff(45,plain,
% 0.20/0.45 (((singleton(empty_set) = singleton(empty_set)) | (~(in(tptp_fun_C_0(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_0(singleton(empty_set), empty_set) = empty_set)))) <=> $true),
% 0.20/0.45 inference(transitivity,[status(thm)],[44, 41])).
% 0.20/0.45 tff(46,plain,
% 0.20/0.45 ((~((singleton(empty_set) = singleton(empty_set)) | (~(in(tptp_fun_C_0(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_0(singleton(empty_set), empty_set) = empty_set))))) <=> (~$true)),
% 0.20/0.45 inference(monotonicity,[status(thm)],[45])).
% 0.20/0.45 tff(47,plain,
% 0.20/0.45 ((~((singleton(empty_set) = singleton(empty_set)) | (~(in(tptp_fun_C_0(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_0(singleton(empty_set), empty_set) = empty_set))))) <=> $false),
% 0.20/0.45 inference(transitivity,[status(thm)],[46, 40])).
% 0.20/0.45 tff(48,plain,
% 0.20/0.45 ((~(in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set))) <=> (~(in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set)))),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(49,plain,
% 0.20/0.45 (($false | (in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set))) <=> (in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set))),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(50,plain,
% 0.20/0.45 ((~(singleton(empty_set) = singleton(empty_set))) <=> (~$true)),
% 0.20/0.45 inference(monotonicity,[status(thm)],[43])).
% 0.20/0.45 tff(51,plain,
% 0.20/0.45 ((~(singleton(empty_set) = singleton(empty_set))) <=> $false),
% 0.20/0.45 inference(transitivity,[status(thm)],[50, 40])).
% 0.20/0.45 tff(52,plain,
% 0.20/0.45 (((~(singleton(empty_set) = singleton(empty_set))) | (in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set))) <=> ($false | (in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set)))),
% 0.20/0.45 inference(monotonicity,[status(thm)],[51])).
% 0.20/0.45 tff(53,plain,
% 0.20/0.45 (((~(singleton(empty_set) = singleton(empty_set))) | (in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set))) <=> (in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set))),
% 0.20/0.45 inference(transitivity,[status(thm)],[52, 49])).
% 0.20/0.45 tff(54,plain,
% 0.20/0.45 ((~((~(singleton(empty_set) = singleton(empty_set))) | (in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set)))) <=> (~(in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set)))),
% 0.20/0.45 inference(monotonicity,[status(thm)],[53])).
% 0.20/0.45 tff(55,plain,
% 0.20/0.45 ((~((~(singleton(empty_set) = singleton(empty_set))) | (in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set)))) <=> (~(in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set)))),
% 0.20/0.45 inference(transitivity,[status(thm)],[54, 48])).
% 0.20/0.45 tff(56,plain,
% 0.20/0.45 (((~((~(singleton(empty_set) = singleton(empty_set))) | (in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set)))) | (~((singleton(empty_set) = singleton(empty_set)) | (~(in(tptp_fun_C_0(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_0(singleton(empty_set), empty_set) = empty_set)))))) <=> ((~(in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set))) | $false)),
% 0.20/0.45 inference(monotonicity,[status(thm)],[55, 47])).
% 0.20/0.45 tff(57,plain,
% 0.20/0.45 (((~((~(singleton(empty_set) = singleton(empty_set))) | (in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set)))) | (~((singleton(empty_set) = singleton(empty_set)) | (~(in(tptp_fun_C_0(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_0(singleton(empty_set), empty_set) = empty_set)))))) <=> (~(in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set)))),
% 0.20/0.46 inference(transitivity,[status(thm)],[56, 39])).
% 0.20/0.46 tff(58,plain,
% 0.20/0.46 ((~((~((~(singleton(empty_set) = singleton(empty_set))) | (in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set)))) | (~((singleton(empty_set) = singleton(empty_set)) | (~(in(tptp_fun_C_0(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_0(singleton(empty_set), empty_set) = empty_set))))))) <=> (~(~(in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set))))),
% 0.20/0.46 inference(monotonicity,[status(thm)],[57])).
% 0.20/0.46 tff(59,plain,
% 0.20/0.46 ((~((~((~(singleton(empty_set) = singleton(empty_set))) | (in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set)))) | (~((singleton(empty_set) = singleton(empty_set)) | (~(in(tptp_fun_C_0(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_0(singleton(empty_set), empty_set) = empty_set))))))) <=> (in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set))),
% 0.20/0.46 inference(transitivity,[status(thm)],[58, 38])).
% 0.20/0.46 tff(60,plain,
% 0.20/0.46 (((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (~((~((~(singleton(empty_set) = singleton(empty_set))) | (in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set)))) | (~((singleton(empty_set) = singleton(empty_set)) | (~(in(tptp_fun_C_0(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_0(singleton(empty_set), empty_set) = empty_set)))))))) <=> ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set)))),
% 0.20/0.46 inference(monotonicity,[status(thm)],[59])).
% 0.20/0.46 tff(61,plain,
% 0.20/0.46 (((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (~((~((~(singleton(empty_set) = singleton(empty_set))) | (in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set)))) | (~((singleton(empty_set) = singleton(empty_set)) | (~(in(tptp_fun_C_0(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_0(singleton(empty_set), empty_set) = empty_set)))))))) <=> ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set)))),
% 0.20/0.46 inference(transitivity,[status(thm)],[60, 37])).
% 0.20/0.46 tff(62,plain,
% 0.20/0.46 ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (~((~((~(singleton(empty_set) = singleton(empty_set))) | (in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set)))) | (~((singleton(empty_set) = singleton(empty_set)) | (~(in(tptp_fun_C_0(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_0(singleton(empty_set), empty_set) = empty_set)))))))),
% 0.20/0.46 inference(quant_inst,[status(thm)],[])).
% 0.20/0.46 tff(63,plain,
% 0.20/0.46 ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set))),
% 0.20/0.46 inference(modus_ponens,[status(thm)],[62, 61])).
% 0.20/0.46 tff(64,plain,
% 0.20/0.46 (in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set)),
% 0.20/0.46 inference(unit_resolution,[status(thm)],[63, 36])).
% 0.20/0.46 tff(65,plain,
% 0.20/0.46 ((~(in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set))) | (~in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set))) | (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set)),
% 0.20/0.46 inference(tautology,[status(thm)],[])).
% 0.20/0.46 tff(66,plain,
% 0.20/0.46 ((~in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set))) | (tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set)),
% 0.20/0.46 inference(unit_resolution,[status(thm)],[65, 64])).
% 0.20/0.46 tff(67,plain,
% 0.20/0.46 (~in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set))),
% 0.20/0.46 inference(unit_resolution,[status(thm)],[66, 1])).
% 0.20/0.46 tff(68,assumption,((~((~(singleton(empty_set) = powerset(empty_set))) | (in(empty_set, singleton(empty_set)) <=> subset(empty_set, empty_set)))) | (~((singleton(empty_set) = powerset(empty_set)) | (~(in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> subset(tptp_fun_C_1(singleton(empty_set), empty_set), empty_set)))))), introduced(assumption)).
% 0.20/0.46 tff(69,plain,
% 0.20/0.46 (^[A: $i, B: $i, C: $i] : refl((~((~((~(B = powerset(A))) | (in(C, B) <=> subset(C, A)))) | (~((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A))))))) <=> (~((~((~(B = powerset(A))) | (in(C, B) <=> subset(C, A)))) | (~((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A))))))))),
% 0.20/0.46 inference(bind,[status(th)],[])).
% 0.20/0.46 tff(70,plain,
% 0.20/0.46 (![A: $i, B: $i, C: $i] : (~((~((~(B = powerset(A))) | (in(C, B) <=> subset(C, A)))) | (~((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A))))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(B = powerset(A))) | (in(C, B) <=> subset(C, A)))) | (~((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A)))))))),
% 0.20/0.46 inference(quant_intro,[status(thm)],[69])).
% 0.20/0.46 tff(71,plain,
% 0.20/0.46 (![A: $i, B: $i] : ![C: $i] : (~((~((~(B = powerset(A))) | (in(C, B) <=> subset(C, A)))) | (~((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A))))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(B = powerset(A))) | (in(C, B) <=> subset(C, A)))) | (~((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A)))))))),
% 0.20/0.46 inference(pull_quant,[status(thm)],[])).
% 0.20/0.46 tff(72,plain,
% 0.20/0.46 (^[A: $i, B: $i] : trans(monotonicity(trans(monotonicity(trans(monotonicity(pull_quant(((~(B = powerset(A))) | ![C: $i] : (in(C, B) <=> subset(C, A))) <=> ![C: $i] : ((~(B = powerset(A))) | (in(C, B) <=> subset(C, A)))), ((~((~(B = powerset(A))) | ![C: $i] : (in(C, B) <=> subset(C, A)))) <=> (~![C: $i] : ((~(B = powerset(A))) | (in(C, B) <=> subset(C, A)))))), pull_quant((~![C: $i] : ((~(B = powerset(A))) | (in(C, B) <=> subset(C, A)))) <=> ?[C: $i] : (~((~(B = powerset(A))) | (in(C, B) <=> subset(C, A))))), ((~((~(B = powerset(A))) | ![C: $i] : (in(C, B) <=> subset(C, A)))) <=> ?[C: $i] : (~((~(B = powerset(A))) | (in(C, B) <=> subset(C, A)))))), (((~((~(B = powerset(A))) | ![C: $i] : (in(C, B) <=> subset(C, A)))) | (~((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A)))))) <=> (?[C: $i] : (~((~(B = powerset(A))) | (in(C, B) <=> subset(C, A)))) | (~((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A)))))))), pull_quant((?[C: $i] : (~((~(B = powerset(A))) | (in(C, B) <=> subset(C, A)))) | (~((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A)))))) <=> ?[C: $i] : ((~((~(B = powerset(A))) | (in(C, B) <=> subset(C, A)))) | (~((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A))))))), (((~((~(B = powerset(A))) | ![C: $i] : (in(C, B) <=> subset(C, A)))) | (~((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A)))))) <=> ?[C: $i] : ((~((~(B = powerset(A))) | (in(C, B) <=> subset(C, A)))) | (~((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A)))))))), ((~((~((~(B = powerset(A))) | ![C: $i] : (in(C, B) <=> subset(C, A)))) | (~((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A))))))) <=> (~?[C: $i] : ((~((~(B = powerset(A))) | (in(C, B) <=> subset(C, A)))) | (~((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A))))))))), pull_quant((~?[C: $i] : ((~((~(B = powerset(A))) | (in(C, B) <=> subset(C, A)))) | (~((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A))))))) <=> ![C: $i] : (~((~((~(B = powerset(A))) | (in(C, B) <=> subset(C, A)))) | (~((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A)))))))), ((~((~((~(B = powerset(A))) | ![C: $i] : (in(C, B) <=> subset(C, A)))) | (~((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A))))))) <=> ![C: $i] : (~((~((~(B = powerset(A))) | (in(C, B) <=> subset(C, A)))) | (~((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A)))))))))),
% 0.20/0.46 inference(bind,[status(th)],[])).
% 0.20/0.46 tff(73,plain,
% 0.20/0.46 (![A: $i, B: $i] : (~((~((~(B = powerset(A))) | ![C: $i] : (in(C, B) <=> subset(C, A)))) | (~((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A))))))) <=> ![A: $i, B: $i] : ![C: $i] : (~((~((~(B = powerset(A))) | (in(C, B) <=> subset(C, A)))) | (~((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A)))))))),
% 0.20/0.46 inference(quant_intro,[status(thm)],[72])).
% 0.20/0.46 tff(74,plain,
% 0.20/0.46 (![A: $i, B: $i] : (~((~((~(B = powerset(A))) | ![C: $i] : (in(C, B) <=> subset(C, A)))) | (~((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A))))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(B = powerset(A))) | (in(C, B) <=> subset(C, A)))) | (~((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A)))))))),
% 0.20/0.46 inference(transitivity,[status(thm)],[73, 71])).
% 0.20/0.46 tff(75,plain,
% 0.20/0.46 (![A: $i, B: $i] : (~((~((~(B = powerset(A))) | ![C: $i] : (in(C, B) <=> subset(C, A)))) | (~((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A))))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(B = powerset(A))) | (in(C, B) <=> subset(C, A)))) | (~((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A)))))))),
% 0.20/0.46 inference(transitivity,[status(thm)],[74, 70])).
% 0.20/0.46 tff(76,plain,
% 0.20/0.46 (^[A: $i, B: $i] : rewrite((~((~((~(B = powerset(A))) | ![C: $i] : (in(C, B) <=> subset(C, A)))) | (~((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A))))))) <=> (~((~((~(B = powerset(A))) | ![C: $i] : (in(C, B) <=> subset(C, A)))) | (~((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A))))))))),
% 0.20/0.46 inference(bind,[status(th)],[])).
% 0.20/0.46 tff(77,plain,
% 0.20/0.46 (![A: $i, B: $i] : (~((~((~(B = powerset(A))) | ![C: $i] : (in(C, B) <=> subset(C, A)))) | (~((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A))))))) <=> ![A: $i, B: $i] : (~((~((~(B = powerset(A))) | ![C: $i] : (in(C, B) <=> subset(C, A)))) | (~((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A)))))))),
% 0.20/0.46 inference(quant_intro,[status(thm)],[76])).
% 0.20/0.46 tff(78,plain,
% 0.20/0.46 (![A: $i, B: $i] : (~((~((~(B = powerset(A))) | ![C: $i] : (in(C, B) <=> subset(C, A)))) | (~((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A))))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(B = powerset(A))) | (in(C, B) <=> subset(C, A)))) | (~((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A)))))))),
% 0.20/0.46 inference(transitivity,[status(thm)],[77, 75])).
% 0.20/0.46 tff(79,plain,
% 0.20/0.46 (^[A: $i, B: $i] : trans(monotonicity(rewrite(((~(B = powerset(A))) | ![C: $i] : (in(C, B) <=> subset(C, A))) <=> ((~(B = powerset(A))) | ![C: $i] : (in(C, B) <=> subset(C, A)))), trans(monotonicity(rewrite((~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A))) <=> (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A)))), (((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A)))) <=> ((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A)))))), rewrite(((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A)))) <=> ((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A))))), (((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A)))) <=> ((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A)))))), ((((~(B = powerset(A))) | ![C: $i] : (in(C, B) <=> subset(C, A))) & ((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A))))) <=> (((~(B = powerset(A))) | ![C: $i] : (in(C, B) <=> subset(C, A))) & ((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A))))))), rewrite((((~(B = powerset(A))) | ![C: $i] : (in(C, B) <=> subset(C, A))) & ((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A))))) <=> (~((~((~(B = powerset(A))) | ![C: $i] : (in(C, B) <=> subset(C, A)))) | (~((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A)))))))), ((((~(B = powerset(A))) | ![C: $i] : (in(C, B) <=> subset(C, A))) & ((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A))))) <=> (~((~((~(B = powerset(A))) | ![C: $i] : (in(C, B) <=> subset(C, A)))) | (~((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A)))))))))),
% 0.20/0.46 inference(bind,[status(th)],[])).
% 0.20/0.46 tff(80,plain,
% 0.20/0.46 (![A: $i, B: $i] : (((~(B = powerset(A))) | ![C: $i] : (in(C, B) <=> subset(C, A))) & ((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A))))) <=> ![A: $i, B: $i] : (~((~((~(B = powerset(A))) | ![C: $i] : (in(C, B) <=> subset(C, A)))) | (~((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A)))))))),
% 0.20/0.46 inference(quant_intro,[status(thm)],[79])).
% 0.20/0.46 tff(81,plain,
% 0.20/0.46 (![A: $i, B: $i] : ((B = powerset(A)) <=> ![C: $i] : (in(C, B) <=> subset(C, A))) <=> ![A: $i, B: $i] : ((B = powerset(A)) <=> ![C: $i] : (in(C, B) <=> subset(C, A)))),
% 0.20/0.46 inference(rewrite,[status(thm)],[])).
% 0.20/0.46 tff(82,axiom,(![A: $i, B: $i] : ((B = powerset(A)) <=> ![C: $i] : (in(C, B) <=> subset(C, A)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','d1_zfmisc_1')).
% 0.20/0.46 tff(83,plain,
% 0.20/0.46 (![A: $i, B: $i] : ((B = powerset(A)) <=> ![C: $i] : (in(C, B) <=> subset(C, A)))),
% 0.20/0.46 inference(modus_ponens,[status(thm)],[82, 81])).
% 0.20/0.46 tff(84,plain,(
% 0.20/0.46 ![A: $i, B: $i] : (((~(B = powerset(A))) | ![C: $i] : (in(C, B) <=> subset(C, A))) & ((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A)))))),
% 0.20/0.46 inference(skolemize,[status(sab)],[83])).
% 0.20/0.46 tff(85,plain,
% 0.20/0.46 (![A: $i, B: $i] : (~((~((~(B = powerset(A))) | ![C: $i] : (in(C, B) <=> subset(C, A)))) | (~((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A)))))))),
% 0.20/0.46 inference(modus_ponens,[status(thm)],[84, 80])).
% 0.20/0.46 tff(86,plain,
% 0.20/0.46 (![A: $i, B: $i, C: $i] : (~((~((~(B = powerset(A))) | (in(C, B) <=> subset(C, A)))) | (~((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A)))))))),
% 0.20/0.46 inference(modus_ponens,[status(thm)],[85, 78])).
% 0.20/0.46 tff(87,plain,
% 0.20/0.46 ((~![A: $i, B: $i, C: $i] : (~((~((~(B = powerset(A))) | (in(C, B) <=> subset(C, A)))) | (~((B = powerset(A)) | (~(in(tptp_fun_C_1(B, A), B) <=> subset(tptp_fun_C_1(B, A), A)))))))) | (~((~((~(singleton(empty_set) = powerset(empty_set))) | (in(empty_set, singleton(empty_set)) <=> subset(empty_set, empty_set)))) | (~((singleton(empty_set) = powerset(empty_set)) | (~(in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> subset(tptp_fun_C_1(singleton(empty_set), empty_set), empty_set)))))))),
% 0.20/0.46 inference(quant_inst,[status(thm)],[])).
% 0.20/0.46 tff(88,plain,
% 0.20/0.46 ($false),
% 0.20/0.46 inference(unit_resolution,[status(thm)],[87, 86, 68])).
% 0.20/0.46 tff(89,plain,(~((~((~(singleton(empty_set) = powerset(empty_set))) | (in(empty_set, singleton(empty_set)) <=> subset(empty_set, empty_set)))) | (~((singleton(empty_set) = powerset(empty_set)) | (~(in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> subset(tptp_fun_C_1(singleton(empty_set), empty_set), empty_set))))))), inference(lemma,lemma(discharge,[]))).
% 0.20/0.46 tff(90,plain,
% 0.20/0.46 (((~((~(singleton(empty_set) = powerset(empty_set))) | (in(empty_set, singleton(empty_set)) <=> subset(empty_set, empty_set)))) | (~((singleton(empty_set) = powerset(empty_set)) | (~(in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> subset(tptp_fun_C_1(singleton(empty_set), empty_set), empty_set)))))) | ((singleton(empty_set) = powerset(empty_set)) | (~(in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> subset(tptp_fun_C_1(singleton(empty_set), empty_set), empty_set))))),
% 0.20/0.47 inference(tautology,[status(thm)],[])).
% 0.20/0.47 tff(91,plain,
% 0.20/0.47 ((singleton(empty_set) = powerset(empty_set)) | (~(in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> subset(tptp_fun_C_1(singleton(empty_set), empty_set), empty_set)))),
% 0.20/0.47 inference(unit_resolution,[status(thm)],[90, 89])).
% 0.20/0.47 tff(92,assumption,(singleton(empty_set) = powerset(empty_set)), introduced(assumption)).
% 0.20/0.47 tff(93,plain,
% 0.20/0.47 (powerset(empty_set) = singleton(empty_set)),
% 0.20/0.47 inference(symmetry,[status(thm)],[92])).
% 0.20/0.47 tff(94,plain,
% 0.20/0.47 ((~(powerset(empty_set) = singleton(empty_set))) <=> (~(powerset(empty_set) = singleton(empty_set)))),
% 0.20/0.47 inference(rewrite,[status(thm)],[])).
% 0.20/0.47 tff(95,axiom,(~(powerset(empty_set) = singleton(empty_set))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t1_zfmisc_1')).
% 0.20/0.47 tff(96,plain,
% 0.20/0.47 (~(powerset(empty_set) = singleton(empty_set))),
% 0.20/0.47 inference(modus_ponens,[status(thm)],[95, 94])).
% 0.20/0.47 tff(97,plain,
% 0.20/0.47 ($false),
% 0.20/0.47 inference(unit_resolution,[status(thm)],[96, 93])).
% 0.20/0.47 tff(98,plain,(~(singleton(empty_set) = powerset(empty_set))), inference(lemma,lemma(discharge,[]))).
% 0.20/0.47 tff(99,plain,
% 0.20/0.47 ((~((singleton(empty_set) = powerset(empty_set)) | (~(in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> subset(tptp_fun_C_1(singleton(empty_set), empty_set), empty_set))))) | (singleton(empty_set) = powerset(empty_set)) | (~(in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> subset(tptp_fun_C_1(singleton(empty_set), empty_set), empty_set)))),
% 0.20/0.47 inference(tautology,[status(thm)],[])).
% 0.20/0.47 tff(100,plain,
% 0.20/0.47 ((~((singleton(empty_set) = powerset(empty_set)) | (~(in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> subset(tptp_fun_C_1(singleton(empty_set), empty_set), empty_set))))) | (~(in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> subset(tptp_fun_C_1(singleton(empty_set), empty_set), empty_set)))),
% 0.20/0.47 inference(unit_resolution,[status(thm)],[99, 98])).
% 0.20/0.47 tff(101,plain,
% 0.20/0.47 (~(in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> subset(tptp_fun_C_1(singleton(empty_set), empty_set), empty_set))),
% 0.20/0.47 inference(unit_resolution,[status(thm)],[100, 91])).
% 0.20/0.47 tff(102,plain,
% 0.20/0.47 ((in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> subset(tptp_fun_C_1(singleton(empty_set), empty_set), empty_set)) | in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) | subset(tptp_fun_C_1(singleton(empty_set), empty_set), empty_set)),
% 0.20/0.47 inference(tautology,[status(thm)],[])).
% 0.20/0.47 tff(103,plain,
% 0.20/0.47 (in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) | subset(tptp_fun_C_1(singleton(empty_set), empty_set), empty_set)),
% 0.20/0.47 inference(unit_resolution,[status(thm)],[102, 101])).
% 0.20/0.47 tff(104,plain,
% 0.20/0.47 ($false),
% 0.20/0.47 inference(unit_resolution,[status(thm)],[103, 67, 18])).
% 0.20/0.47 tff(105,plain,(tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set), inference(lemma,lemma(discharge,[]))).
% 0.20/0.47 tff(106,plain,
% 0.20/0.47 (subset(tptp_fun_C_1(singleton(empty_set), empty_set), empty_set) <=> subset(empty_set, empty_set)),
% 0.20/0.47 inference(monotonicity,[status(thm)],[105])).
% 0.20/0.47 tff(107,plain,
% 0.20/0.47 (subset(empty_set, empty_set) <=> subset(tptp_fun_C_1(singleton(empty_set), empty_set), empty_set)),
% 0.20/0.47 inference(symmetry,[status(thm)],[106])).
% 0.20/0.47 tff(108,assumption,(~subset(empty_set, empty_set)), introduced(assumption)).
% 0.20/0.47 tff(109,plain,
% 0.20/0.47 (^[A: $i] : refl(subset(A, A) <=> subset(A, A))),
% 0.20/0.47 inference(bind,[status(th)],[])).
% 0.20/0.47 tff(110,plain,
% 0.20/0.47 (![A: $i] : subset(A, A) <=> ![A: $i] : subset(A, A)),
% 0.20/0.47 inference(quant_intro,[status(thm)],[109])).
% 0.20/0.47 tff(111,plain,
% 0.20/0.47 (![A: $i] : subset(A, A) <=> ![A: $i] : subset(A, A)),
% 0.20/0.47 inference(rewrite,[status(thm)],[])).
% 0.20/0.47 tff(112,plain,
% 0.20/0.47 (![A: $i, B: $i] : subset(A, A) <=> ![A: $i] : subset(A, A)),
% 0.20/0.47 inference(elim_unused_vars,[status(thm)],[])).
% 0.20/0.47 tff(113,axiom,(![A: $i, B: $i] : subset(A, A)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','reflexivity_r1_tarski')).
% 0.20/0.47 tff(114,plain,
% 0.20/0.47 (![A: $i] : subset(A, A)),
% 0.20/0.47 inference(modus_ponens,[status(thm)],[113, 112])).
% 0.20/0.47 tff(115,plain,
% 0.20/0.47 (![A: $i] : subset(A, A)),
% 0.20/0.47 inference(modus_ponens,[status(thm)],[114, 111])).
% 0.20/0.47 tff(116,plain,(
% 0.20/0.47 ![A: $i] : subset(A, A)),
% 0.20/0.47 inference(skolemize,[status(sab)],[115])).
% 0.20/0.47 tff(117,plain,
% 0.20/0.47 (![A: $i] : subset(A, A)),
% 0.20/0.47 inference(modus_ponens,[status(thm)],[116, 110])).
% 0.20/0.47 tff(118,plain,
% 0.20/0.47 ((~![A: $i] : subset(A, A)) | subset(empty_set, empty_set)),
% 0.20/0.47 inference(quant_inst,[status(thm)],[])).
% 0.20/0.47 tff(119,plain,
% 0.20/0.47 ($false),
% 0.20/0.47 inference(unit_resolution,[status(thm)],[118, 117, 108])).
% 0.20/0.47 tff(120,plain,(subset(empty_set, empty_set)), inference(lemma,lemma(discharge,[]))).
% 0.20/0.47 tff(121,plain,
% 0.20/0.47 (subset(tptp_fun_C_1(singleton(empty_set), empty_set), empty_set)),
% 0.20/0.47 inference(modus_ponens,[status(thm)],[120, 107])).
% 0.20/0.47 tff(122,assumption,(tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set), introduced(assumption)).
% 0.20/0.47 tff(123,plain,
% 0.20/0.47 (empty_set = tptp_fun_C_1(singleton(empty_set), empty_set)),
% 0.20/0.47 inference(symmetry,[status(thm)],[122])).
% 0.20/0.47 tff(124,plain,
% 0.20/0.47 (in(empty_set, singleton(empty_set)) <=> in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set))),
% 0.20/0.47 inference(monotonicity,[status(thm)],[123])).
% 0.20/0.47 tff(125,plain,
% 0.20/0.47 (in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> in(empty_set, singleton(empty_set))),
% 0.20/0.47 inference(symmetry,[status(thm)],[124])).
% 0.20/0.47 tff(126,plain,
% 0.20/0.47 ((~in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set))) <=> (~in(empty_set, singleton(empty_set)))),
% 0.20/0.47 inference(monotonicity,[status(thm)],[125])).
% 0.20/0.47 tff(127,assumption,(~in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set))), introduced(assumption)).
% 0.20/0.47 tff(128,plain,
% 0.20/0.47 (~in(empty_set, singleton(empty_set))),
% 0.20/0.47 inference(modus_ponens,[status(thm)],[127, 126])).
% 0.20/0.47 tff(129,plain,
% 0.20/0.47 (((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | in(empty_set, singleton(empty_set))) <=> ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | in(empty_set, singleton(empty_set)))),
% 0.20/0.47 inference(rewrite,[status(thm)],[])).
% 0.20/0.47 tff(130,plain,
% 0.20/0.47 ((~(~in(empty_set, singleton(empty_set)))) <=> in(empty_set, singleton(empty_set))),
% 0.20/0.47 inference(rewrite,[status(thm)],[])).
% 0.20/0.47 tff(131,plain,
% 0.20/0.47 (((~in(empty_set, singleton(empty_set))) | $false) <=> (~in(empty_set, singleton(empty_set)))),
% 0.20/0.47 inference(rewrite,[status(thm)],[])).
% 0.20/0.47 tff(132,plain,
% 0.20/0.47 ((~((~(singleton(empty_set) = singleton(empty_set))) | (in(empty_set, singleton(empty_set)) <=> (empty_set = empty_set)))) <=> (~in(empty_set, singleton(empty_set)))),
% 0.20/0.47 inference(rewrite,[status(thm)],[])).
% 0.20/0.47 tff(133,plain,
% 0.20/0.47 (((~((~(singleton(empty_set) = singleton(empty_set))) | (in(empty_set, singleton(empty_set)) <=> (empty_set = empty_set)))) | (~((singleton(empty_set) = singleton(empty_set)) | (~(in(tptp_fun_C_0(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_0(singleton(empty_set), empty_set) = empty_set)))))) <=> ((~in(empty_set, singleton(empty_set))) | $false)),
% 0.20/0.47 inference(monotonicity,[status(thm)],[132, 47])).
% 0.20/0.47 tff(134,plain,
% 0.20/0.47 (((~((~(singleton(empty_set) = singleton(empty_set))) | (in(empty_set, singleton(empty_set)) <=> (empty_set = empty_set)))) | (~((singleton(empty_set) = singleton(empty_set)) | (~(in(tptp_fun_C_0(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_0(singleton(empty_set), empty_set) = empty_set)))))) <=> (~in(empty_set, singleton(empty_set)))),
% 0.20/0.47 inference(transitivity,[status(thm)],[133, 131])).
% 0.20/0.47 tff(135,plain,
% 0.20/0.47 ((~((~((~(singleton(empty_set) = singleton(empty_set))) | (in(empty_set, singleton(empty_set)) <=> (empty_set = empty_set)))) | (~((singleton(empty_set) = singleton(empty_set)) | (~(in(tptp_fun_C_0(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_0(singleton(empty_set), empty_set) = empty_set))))))) <=> (~(~in(empty_set, singleton(empty_set))))),
% 0.20/0.47 inference(monotonicity,[status(thm)],[134])).
% 0.20/0.47 tff(136,plain,
% 0.20/0.47 ((~((~((~(singleton(empty_set) = singleton(empty_set))) | (in(empty_set, singleton(empty_set)) <=> (empty_set = empty_set)))) | (~((singleton(empty_set) = singleton(empty_set)) | (~(in(tptp_fun_C_0(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_0(singleton(empty_set), empty_set) = empty_set))))))) <=> in(empty_set, singleton(empty_set))),
% 0.20/0.47 inference(transitivity,[status(thm)],[135, 130])).
% 0.20/0.47 tff(137,plain,
% 0.20/0.47 (((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (~((~((~(singleton(empty_set) = singleton(empty_set))) | (in(empty_set, singleton(empty_set)) <=> (empty_set = empty_set)))) | (~((singleton(empty_set) = singleton(empty_set)) | (~(in(tptp_fun_C_0(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_0(singleton(empty_set), empty_set) = empty_set)))))))) <=> ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | in(empty_set, singleton(empty_set)))),
% 0.20/0.47 inference(monotonicity,[status(thm)],[136])).
% 0.20/0.47 tff(138,plain,
% 0.20/0.47 (((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (~((~((~(singleton(empty_set) = singleton(empty_set))) | (in(empty_set, singleton(empty_set)) <=> (empty_set = empty_set)))) | (~((singleton(empty_set) = singleton(empty_set)) | (~(in(tptp_fun_C_0(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_0(singleton(empty_set), empty_set) = empty_set)))))))) <=> ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | in(empty_set, singleton(empty_set)))),
% 0.20/0.47 inference(transitivity,[status(thm)],[137, 129])).
% 0.20/0.47 tff(139,plain,
% 0.20/0.47 ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (~((~((~(singleton(empty_set) = singleton(empty_set))) | (in(empty_set, singleton(empty_set)) <=> (empty_set = empty_set)))) | (~((singleton(empty_set) = singleton(empty_set)) | (~(in(tptp_fun_C_0(singleton(empty_set), empty_set), singleton(empty_set)) <=> (tptp_fun_C_0(singleton(empty_set), empty_set) = empty_set)))))))),
% 0.20/0.47 inference(quant_inst,[status(thm)],[])).
% 0.20/0.47 tff(140,plain,
% 0.20/0.47 ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | in(empty_set, singleton(empty_set))),
% 0.20/0.47 inference(modus_ponens,[status(thm)],[139, 138])).
% 0.20/0.47 tff(141,plain,
% 0.20/0.47 (in(empty_set, singleton(empty_set))),
% 0.20/0.47 inference(unit_resolution,[status(thm)],[140, 36])).
% 0.20/0.47 tff(142,plain,
% 0.20/0.47 ($false),
% 0.20/0.47 inference(unit_resolution,[status(thm)],[141, 128])).
% 0.20/0.47 tff(143,plain,(in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) | (~(tptp_fun_C_1(singleton(empty_set), empty_set) = empty_set))), inference(lemma,lemma(discharge,[]))).
% 0.20/0.47 tff(144,plain,
% 0.20/0.47 (in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set))),
% 0.20/0.47 inference(unit_resolution,[status(thm)],[143, 105])).
% 0.20/0.47 tff(145,plain,
% 0.20/0.47 ((in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set)) <=> subset(tptp_fun_C_1(singleton(empty_set), empty_set), empty_set)) | (~in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set))) | (~subset(tptp_fun_C_1(singleton(empty_set), empty_set), empty_set))),
% 0.20/0.47 inference(tautology,[status(thm)],[])).
% 0.20/0.47 tff(146,plain,
% 0.20/0.47 ((~in(tptp_fun_C_1(singleton(empty_set), empty_set), singleton(empty_set))) | (~subset(tptp_fun_C_1(singleton(empty_set), empty_set), empty_set))),
% 0.20/0.47 inference(unit_resolution,[status(thm)],[145, 101])).
% 0.20/0.47 tff(147,plain,
% 0.20/0.47 (~subset(tptp_fun_C_1(singleton(empty_set), empty_set), empty_set)),
% 0.20/0.47 inference(unit_resolution,[status(thm)],[146, 144])).
% 0.20/0.47 tff(148,plain,
% 0.20/0.47 ($false),
% 0.20/0.47 inference(unit_resolution,[status(thm)],[147, 121])).
% 0.20/0.47 % SZS output end Proof
% 0.20/0.47 % E exiting
%------------------------------------------------------------------------------