%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET907+1 : TPTP v9.0.0. Released v3.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n009.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 05:26:38 AM UTC 2025
% Result : Theorem 0.11s 0.35s
% Output : Proof 0.11s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.05/0.11 % Problem : SET907+1 : TPTP v9.0.0. Released v3.2.0.
% 0.05/0.11 % Command : run_E %s %d THM
% 0.11/0.32 % Computer : n009.cluster.edu
% 0.11/0.32 % Model : x86_64 x86_64
% 0.11/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.32 % Memory : 8042.1875MB
% 0.11/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.32 % CPULimit : 300
% 0.11/0.32 % WCLimit : 300
% 0.11/0.32 % DateTime : Fri Jun 20 05:50:18 EDT 2025
% 0.11/0.32 % CPUTime :
% 0.11/0.35 % SZS status Theorem
% 0.11/0.35 % SZS output start Proof
% 0.11/0.35 tff(subset_type, type, (
% 0.11/0.35 subset: ( $i * $i ) > $o)).
% 0.11/0.35 tff(tptp_fun_B_3_type, type, (
% 0.11/0.35 tptp_fun_B_3: $i)).
% 0.11/0.35 tff(unordered_pair_type, type, (
% 0.11/0.35 unordered_pair: ( $i * $i ) > $i)).
% 0.11/0.35 tff(tptp_fun_C_2_type, type, (
% 0.11/0.35 tptp_fun_C_2: $i)).
% 0.11/0.35 tff(tptp_fun_A_4_type, type, (
% 0.11/0.35 tptp_fun_A_4: $i)).
% 0.11/0.35 tff(in_type, type, (
% 0.11/0.35 in: ( $i * $i ) > $o)).
% 0.11/0.35 tff(set_union2_type, type, (
% 0.11/0.35 set_union2: ( $i * $i ) > $i)).
% 0.11/0.35 tff(1,plain,
% 0.11/0.35 ((~![A: $i, B: $i, C: $i] : ((~(in(A, B) & in(C, B))) | (set_union2(unordered_pair(A, C), B) = B))) <=> (~![A: $i, B: $i, C: $i] : ((~(in(A, B) & in(C, B))) | (set_union2(unordered_pair(A, C), B) = B)))),
% 0.11/0.35 inference(rewrite,[status(thm)],[])).
% 0.11/0.35 tff(2,plain,
% 0.11/0.35 ((~![A: $i, B: $i, C: $i] : ((in(A, B) & in(C, B)) => (set_union2(unordered_pair(A, C), B) = B))) <=> (~![A: $i, B: $i, C: $i] : ((~(in(A, B) & in(C, B))) | (set_union2(unordered_pair(A, C), B) = B)))),
% 0.11/0.35 inference(rewrite,[status(thm)],[])).
% 0.11/0.35 tff(3,axiom,(~![A: $i, B: $i, C: $i] : ((in(A, B) & in(C, B)) => (set_union2(unordered_pair(A, C), B) = B))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t48_zfmisc_1')).
% 0.11/0.35 tff(4,plain,
% 0.11/0.35 (~![A: $i, B: $i, C: $i] : ((~(in(A, B) & in(C, B))) | (set_union2(unordered_pair(A, C), B) = B))),
% 0.11/0.35 inference(modus_ponens,[status(thm)],[3, 2])).
% 0.11/0.35 tff(5,plain,
% 0.11/0.35 (~![A: $i, B: $i, C: $i] : ((~(in(A, B) & in(C, B))) | (set_union2(unordered_pair(A, C), B) = B))),
% 0.11/0.35 inference(modus_ponens,[status(thm)],[4, 1])).
% 0.11/0.35 tff(6,plain,
% 0.11/0.35 (~![A: $i, B: $i, C: $i] : ((~(in(A, B) & in(C, B))) | (set_union2(unordered_pair(A, C), B) = B))),
% 0.11/0.35 inference(modus_ponens,[status(thm)],[5, 1])).
% 0.11/0.35 tff(7,plain,
% 0.11/0.35 (~![A: $i, B: $i, C: $i] : ((~(in(A, B) & in(C, B))) | (set_union2(unordered_pair(A, C), B) = B))),
% 0.11/0.35 inference(modus_ponens,[status(thm)],[6, 1])).
% 0.11/0.35 tff(8,plain,(
% 0.11/0.35 ~((~(in(A!4, B!3) & in(C!2, B!3))) | (set_union2(unordered_pair(A!4, C!2), B!3) = B!3))),
% 0.11/0.35 inference(skolemize,[status(sab)],[7])).
% 0.11/0.35 tff(9,plain,
% 0.11/0.35 (in(A!4, B!3) & in(C!2, B!3)),
% 0.11/0.35 inference(or_elim,[status(thm)],[8])).
% 0.11/0.35 tff(10,plain,
% 0.11/0.35 (in(C!2, B!3)),
% 0.11/0.35 inference(and_elim,[status(thm)],[9])).
% 0.11/0.35 tff(11,plain,
% 0.11/0.35 (in(A!4, B!3)),
% 0.11/0.35 inference(and_elim,[status(thm)],[9])).
% 0.11/0.35 tff(12,plain,
% 0.11/0.35 ((~((~in(A!4, B!3)) | (~in(C!2, B!3)))) | (~in(A!4, B!3)) | (~in(C!2, B!3))),
% 0.11/0.35 inference(tautology,[status(thm)],[])).
% 0.11/0.35 tff(13,plain,
% 0.11/0.35 (~((~in(A!4, B!3)) | (~in(C!2, B!3)))),
% 0.11/0.35 inference(unit_resolution,[status(thm)],[12, 11, 10])).
% 0.11/0.35 tff(14,plain,
% 0.11/0.35 (~(set_union2(unordered_pair(A!4, C!2), B!3) = B!3)),
% 0.11/0.35 inference(or_elim,[status(thm)],[8])).
% 0.11/0.35 tff(15,plain,
% 0.11/0.35 (^[A: $i, B: $i] : refl(((~subset(A, B)) | (set_union2(A, B) = B)) <=> ((~subset(A, B)) | (set_union2(A, B) = B)))),
% 0.11/0.35 inference(bind,[status(th)],[])).
% 0.11/0.35 tff(16,plain,
% 0.11/0.35 (![A: $i, B: $i] : ((~subset(A, B)) | (set_union2(A, B) = B)) <=> ![A: $i, B: $i] : ((~subset(A, B)) | (set_union2(A, B) = B))),
% 0.11/0.35 inference(quant_intro,[status(thm)],[15])).
% 0.11/0.35 tff(17,plain,
% 0.11/0.35 (![A: $i, B: $i] : ((~subset(A, B)) | (set_union2(A, B) = B)) <=> ![A: $i, B: $i] : ((~subset(A, B)) | (set_union2(A, B) = B))),
% 0.11/0.36 inference(rewrite,[status(thm)],[])).
% 0.11/0.36 tff(18,plain,
% 0.11/0.36 (^[A: $i, B: $i] : rewrite((subset(A, B) => (set_union2(A, B) = B)) <=> ((~subset(A, B)) | (set_union2(A, B) = B)))),
% 0.11/0.36 inference(bind,[status(th)],[])).
% 0.11/0.36 tff(19,plain,
% 0.11/0.36 (![A: $i, B: $i] : (subset(A, B) => (set_union2(A, B) = B)) <=> ![A: $i, B: $i] : ((~subset(A, B)) | (set_union2(A, B) = B))),
% 0.11/0.36 inference(quant_intro,[status(thm)],[18])).
% 0.11/0.36 tff(20,axiom,(![A: $i, B: $i] : (subset(A, B) => (set_union2(A, B) = B))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t12_xboole_1')).
% 0.11/0.36 tff(21,plain,
% 0.11/0.36 (![A: $i, B: $i] : ((~subset(A, B)) | (set_union2(A, B) = B))),
% 0.11/0.36 inference(modus_ponens,[status(thm)],[20, 19])).
% 0.11/0.36 tff(22,plain,
% 0.11/0.36 (![A: $i, B: $i] : ((~subset(A, B)) | (set_union2(A, B) = B))),
% 0.11/0.36 inference(modus_ponens,[status(thm)],[21, 17])).
% 0.11/0.36 tff(23,plain,(
% 0.11/0.36 ![A: $i, B: $i] : ((~subset(A, B)) | (set_union2(A, B) = B))),
% 0.11/0.36 inference(skolemize,[status(sab)],[22])).
% 0.11/0.36 tff(24,plain,
% 0.11/0.36 (![A: $i, B: $i] : ((~subset(A, B)) | (set_union2(A, B) = B))),
% 0.11/0.36 inference(modus_ponens,[status(thm)],[23, 16])).
% 0.11/0.36 tff(25,plain,
% 0.11/0.36 (((~![A: $i, B: $i] : ((~subset(A, B)) | (set_union2(A, B) = B))) | ((~subset(unordered_pair(A!4, C!2), B!3)) | (set_union2(unordered_pair(A!4, C!2), B!3) = B!3))) <=> ((~![A: $i, B: $i] : ((~subset(A, B)) | (set_union2(A, B) = B))) | (~subset(unordered_pair(A!4, C!2), B!3)) | (set_union2(unordered_pair(A!4, C!2), B!3) = B!3))),
% 0.11/0.36 inference(rewrite,[status(thm)],[])).
% 0.11/0.36 tff(26,plain,
% 0.11/0.36 ((~![A: $i, B: $i] : ((~subset(A, B)) | (set_union2(A, B) = B))) | ((~subset(unordered_pair(A!4, C!2), B!3)) | (set_union2(unordered_pair(A!4, C!2), B!3) = B!3))),
% 0.11/0.36 inference(quant_inst,[status(thm)],[])).
% 0.11/0.36 tff(27,plain,
% 0.11/0.36 ((~![A: $i, B: $i] : ((~subset(A, B)) | (set_union2(A, B) = B))) | (~subset(unordered_pair(A!4, C!2), B!3)) | (set_union2(unordered_pair(A!4, C!2), B!3) = B!3)),
% 0.11/0.36 inference(modus_ponens,[status(thm)],[26, 25])).
% 0.11/0.36 tff(28,plain,
% 0.11/0.36 (~subset(unordered_pair(A!4, C!2), B!3)),
% 0.11/0.36 inference(unit_resolution,[status(thm)],[27, 24, 14])).
% 0.11/0.36 tff(29,plain,
% 0.11/0.36 ((((~in(A!4, B!3)) | (~in(C!2, B!3))) <=> subset(unordered_pair(A!4, C!2), B!3)) | ((~in(A!4, B!3)) | (~in(C!2, B!3))) | subset(unordered_pair(A!4, C!2), B!3)),
% 0.11/0.36 inference(tautology,[status(thm)],[])).
% 0.11/0.36 tff(30,plain,
% 0.11/0.36 (((~in(A!4, B!3)) | (~in(C!2, B!3))) <=> subset(unordered_pair(A!4, C!2), B!3)),
% 0.11/0.36 inference(unit_resolution,[status(thm)],[29, 28, 13])).
% 0.11/0.36 tff(31,plain,
% 0.11/0.36 (^[A: $i, B: $i, C: $i] : refl((~(((~in(B, C)) | (~in(A, C))) <=> subset(unordered_pair(A, B), C))) <=> (~(((~in(B, C)) | (~in(A, C))) <=> subset(unordered_pair(A, B), C))))),
% 0.11/0.36 inference(bind,[status(th)],[])).
% 0.11/0.36 tff(32,plain,
% 0.11/0.36 (![A: $i, B: $i, C: $i] : (~(((~in(B, C)) | (~in(A, C))) <=> subset(unordered_pair(A, B), C))) <=> ![A: $i, B: $i, C: $i] : (~(((~in(B, C)) | (~in(A, C))) <=> subset(unordered_pair(A, B), C)))),
% 0.11/0.36 inference(quant_intro,[status(thm)],[31])).
% 0.11/0.36 tff(33,plain,
% 0.11/0.36 (^[A: $i, B: $i, C: $i] : trans(trans(monotonicity(rewrite((in(A, C) & in(B, C)) <=> (~((~in(B, C)) | (~in(A, C))))), ((subset(unordered_pair(A, B), C) <=> (in(A, C) & in(B, C))) <=> (subset(unordered_pair(A, B), C) <=> (~((~in(B, C)) | (~in(A, C))))))), rewrite((subset(unordered_pair(A, B), C) <=> (~((~in(B, C)) | (~in(A, C))))) <=> (~(((~in(B, C)) | (~in(A, C))) <=> subset(unordered_pair(A, B), C)))), ((subset(unordered_pair(A, B), C) <=> (in(A, C) & in(B, C))) <=> (~(((~in(B, C)) | (~in(A, C))) <=> subset(unordered_pair(A, B), C))))), rewrite((~(((~in(B, C)) | (~in(A, C))) <=> subset(unordered_pair(A, B), C))) <=> (~(((~in(B, C)) | (~in(A, C))) <=> subset(unordered_pair(A, B), C)))), ((subset(unordered_pair(A, B), C) <=> (in(A, C) & in(B, C))) <=> (~(((~in(B, C)) | (~in(A, C))) <=> subset(unordered_pair(A, B), C)))))),
% 0.11/0.36 inference(bind,[status(th)],[])).
% 0.11/0.36 tff(34,plain,
% 0.11/0.36 (![A: $i, B: $i, C: $i] : (subset(unordered_pair(A, B), C) <=> (in(A, C) & in(B, C))) <=> ![A: $i, B: $i, C: $i] : (~(((~in(B, C)) | (~in(A, C))) <=> subset(unordered_pair(A, B), C)))),
% 0.11/0.36 inference(quant_intro,[status(thm)],[33])).
% 0.11/0.36 tff(35,plain,
% 0.11/0.36 (![A: $i, B: $i, C: $i] : (subset(unordered_pair(A, B), C) <=> (in(A, C) & in(B, C))) <=> ![A: $i, B: $i, C: $i] : (subset(unordered_pair(A, B), C) <=> (in(A, C) & in(B, C)))),
% 0.11/0.36 inference(rewrite,[status(thm)],[])).
% 0.11/0.36 tff(36,axiom,(![A: $i, B: $i, C: $i] : (subset(unordered_pair(A, B), C) <=> (in(A, C) & in(B, C)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t38_zfmisc_1')).
% 0.11/0.36 tff(37,plain,
% 0.11/0.36 (![A: $i, B: $i, C: $i] : (subset(unordered_pair(A, B), C) <=> (in(A, C) & in(B, C)))),
% 0.11/0.36 inference(modus_ponens,[status(thm)],[36, 35])).
% 0.11/0.36 tff(38,plain,(
% 0.11/0.36 ![A: $i, B: $i, C: $i] : (subset(unordered_pair(A, B), C) <=> (in(A, C) & in(B, C)))),
% 0.11/0.36 inference(skolemize,[status(sab)],[37])).
% 0.11/0.36 tff(39,plain,
% 0.11/0.36 (![A: $i, B: $i, C: $i] : (~(((~in(B, C)) | (~in(A, C))) <=> subset(unordered_pair(A, B), C)))),
% 0.11/0.36 inference(modus_ponens,[status(thm)],[38, 34])).
% 0.11/0.36 tff(40,plain,
% 0.11/0.36 (![A: $i, B: $i, C: $i] : (~(((~in(B, C)) | (~in(A, C))) <=> subset(unordered_pair(A, B), C)))),
% 0.11/0.36 inference(modus_ponens,[status(thm)],[39, 32])).
% 0.11/0.36 tff(41,plain,
% 0.11/0.36 (((~![A: $i, B: $i, C: $i] : (~(((~in(B, C)) | (~in(A, C))) <=> subset(unordered_pair(A, B), C)))) | (~(((~in(A!4, B!3)) | (~in(C!2, B!3))) <=> subset(unordered_pair(A!4, C!2), B!3)))) <=> ((~![A: $i, B: $i, C: $i] : (~(((~in(B, C)) | (~in(A, C))) <=> subset(unordered_pair(A, B), C)))) | (~(((~in(A!4, B!3)) | (~in(C!2, B!3))) <=> subset(unordered_pair(A!4, C!2), B!3))))),
% 0.11/0.36 inference(rewrite,[status(thm)],[])).
% 0.11/0.36 tff(42,plain,
% 0.11/0.36 ((~(((~in(A!4, B!3)) | (~in(C!2, B!3))) <=> subset(unordered_pair(A!4, C!2), B!3))) <=> (~(((~in(A!4, B!3)) | (~in(C!2, B!3))) <=> subset(unordered_pair(A!4, C!2), B!3)))),
% 0.11/0.36 inference(rewrite,[status(thm)],[])).
% 0.11/0.36 tff(43,plain,
% 0.11/0.36 ((((~in(C!2, B!3)) | (~in(A!4, B!3))) <=> subset(unordered_pair(A!4, C!2), B!3)) <=> (((~in(A!4, B!3)) | (~in(C!2, B!3))) <=> subset(unordered_pair(A!4, C!2), B!3))),
% 0.11/0.36 inference(rewrite,[status(thm)],[])).
% 0.11/0.36 tff(44,plain,
% 0.11/0.36 ((~(((~in(C!2, B!3)) | (~in(A!4, B!3))) <=> subset(unordered_pair(A!4, C!2), B!3))) <=> (~(((~in(A!4, B!3)) | (~in(C!2, B!3))) <=> subset(unordered_pair(A!4, C!2), B!3)))),
% 0.11/0.36 inference(monotonicity,[status(thm)],[43])).
% 0.11/0.36 tff(45,plain,
% 0.11/0.36 ((~(((~in(C!2, B!3)) | (~in(A!4, B!3))) <=> subset(unordered_pair(A!4, C!2), B!3))) <=> (~(((~in(A!4, B!3)) | (~in(C!2, B!3))) <=> subset(unordered_pair(A!4, C!2), B!3)))),
% 0.11/0.36 inference(transitivity,[status(thm)],[44, 42])).
% 0.11/0.36 tff(46,plain,
% 0.11/0.36 (((~![A: $i, B: $i, C: $i] : (~(((~in(B, C)) | (~in(A, C))) <=> subset(unordered_pair(A, B), C)))) | (~(((~in(C!2, B!3)) | (~in(A!4, B!3))) <=> subset(unordered_pair(A!4, C!2), B!3)))) <=> ((~![A: $i, B: $i, C: $i] : (~(((~in(B, C)) | (~in(A, C))) <=> subset(unordered_pair(A, B), C)))) | (~(((~in(A!4, B!3)) | (~in(C!2, B!3))) <=> subset(unordered_pair(A!4, C!2), B!3))))),
% 0.11/0.36 inference(monotonicity,[status(thm)],[45])).
% 0.11/0.36 tff(47,plain,
% 0.11/0.36 (((~![A: $i, B: $i, C: $i] : (~(((~in(B, C)) | (~in(A, C))) <=> subset(unordered_pair(A, B), C)))) | (~(((~in(C!2, B!3)) | (~in(A!4, B!3))) <=> subset(unordered_pair(A!4, C!2), B!3)))) <=> ((~![A: $i, B: $i, C: $i] : (~(((~in(B, C)) | (~in(A, C))) <=> subset(unordered_pair(A, B), C)))) | (~(((~in(A!4, B!3)) | (~in(C!2, B!3))) <=> subset(unordered_pair(A!4, C!2), B!3))))),
% 0.11/0.36 inference(transitivity,[status(thm)],[46, 41])).
% 0.11/0.36 tff(48,plain,
% 0.11/0.36 ((~![A: $i, B: $i, C: $i] : (~(((~in(B, C)) | (~in(A, C))) <=> subset(unordered_pair(A, B), C)))) | (~(((~in(C!2, B!3)) | (~in(A!4, B!3))) <=> subset(unordered_pair(A!4, C!2), B!3)))),
% 0.11/0.36 inference(quant_inst,[status(thm)],[])).
% 0.11/0.36 tff(49,plain,
% 0.11/0.36 ((~![A: $i, B: $i, C: $i] : (~(((~in(B, C)) | (~in(A, C))) <=> subset(unordered_pair(A, B), C)))) | (~(((~in(A!4, B!3)) | (~in(C!2, B!3))) <=> subset(unordered_pair(A!4, C!2), B!3)))),
% 0.11/0.36 inference(modus_ponens,[status(thm)],[48, 47])).
% 0.11/0.36 tff(50,plain,
% 0.11/0.36 ($false),
% 0.11/0.36 inference(unit_resolution,[status(thm)],[49, 40, 30])).
% 0.11/0.36 % SZS output end Proof
% 0.11/0.36 % E exiting
%------------------------------------------------------------------------------