%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SEU326+1 : TPTP v9.0.0. Released v3.3.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n001.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 05:27:36 AM UTC 2025
% Result : Theorem 0.80s 1.00s
% Output : Proof 0.87s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.13/0.13 % Problem : SEU326+1 : TPTP v9.0.0. Released v3.3.0.
% 0.13/0.13 % Command : run_E %s %d THM
% 0.14/0.35 % Computer : n001.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Fri Jun 20 06:49:20 EDT 2025
% 0.14/0.35 % CPUTime :
% 0.80/1.00 % SZS status Theorem
% 0.80/1.00 % SZS output start Proof
% 0.80/1.00 tff(empty_set_type, type, (
% 0.80/1.00 empty_set: $i)).
% 0.80/1.00 tff(tptp_fun_B_4_type, type, (
% 0.80/1.00 tptp_fun_B_4: $i)).
% 0.80/1.00 tff(complements_of_subsets_type, type, (
% 0.80/1.00 complements_of_subsets: ( $i * $i ) > $i)).
% 0.80/1.00 tff(tptp_fun_A_5_type, type, (
% 0.80/1.00 tptp_fun_A_5: $i)).
% 0.80/1.00 tff(element_type, type, (
% 0.80/1.00 element: ( $i * $i ) > $o)).
% 0.80/1.00 tff(powerset_type, type, (
% 0.80/1.00 powerset: $i > $i)).
% 0.80/1.00 tff(1,plain,
% 0.80/1.00 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) & (~((~(complements_of_subsets(A, B) = empty_set)) & (B = empty_set)))))) <=> (~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) & (~((~(complements_of_subsets(A, B) = empty_set)) & (B = empty_set))))))),
% 0.80/1.00 inference(rewrite,[status(thm)],[])).
% 0.80/1.00 tff(2,plain,
% 0.80/1.00 ((~![A: $i, B: $i] : (element(B, powerset(powerset(A))) => ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) & (~((~(complements_of_subsets(A, B) = empty_set)) & (B = empty_set)))))) <=> (~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) & (~((~(complements_of_subsets(A, B) = empty_set)) & (B = empty_set))))))),
% 0.80/1.00 inference(rewrite,[status(thm)],[])).
% 0.80/1.00 tff(3,axiom,(~![A: $i, B: $i] : (element(B, powerset(powerset(A))) => ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) & (~((~(complements_of_subsets(A, B) = empty_set)) & (B = empty_set)))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t10_tops_2')).
% 0.80/1.00 tff(4,plain,
% 0.80/1.00 (~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) & (~((~(complements_of_subsets(A, B) = empty_set)) & (B = empty_set)))))),
% 0.80/1.00 inference(modus_ponens,[status(thm)],[3, 2])).
% 0.80/1.00 tff(5,plain,
% 0.80/1.00 (~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) & (~((~(complements_of_subsets(A, B) = empty_set)) & (B = empty_set)))))),
% 0.80/1.00 inference(modus_ponens,[status(thm)],[4, 1])).
% 0.80/1.00 tff(6,plain,
% 0.80/1.00 (~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) & (~((~(complements_of_subsets(A, B) = empty_set)) & (B = empty_set)))))),
% 0.80/1.00 inference(modus_ponens,[status(thm)],[5, 1])).
% 0.80/1.00 tff(7,plain,
% 0.80/1.00 (~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) & (~((~(complements_of_subsets(A, B) = empty_set)) & (B = empty_set)))))),
% 0.80/1.00 inference(modus_ponens,[status(thm)],[6, 1])).
% 0.80/1.00 tff(8,plain,(
% 0.80/1.00 ~((~element(B!4, powerset(powerset(A!5)))) | ((~((~(B!4 = empty_set)) & (complements_of_subsets(A!5, B!4) = empty_set))) & (~((~(complements_of_subsets(A!5, B!4) = empty_set)) & (B!4 = empty_set)))))),
% 0.80/1.00 inference(skolemize,[status(sab)],[7])).
% 0.80/1.00 tff(9,plain,
% 0.80/1.00 (element(B!4, powerset(powerset(A!5)))),
% 0.80/1.00 inference(or_elim,[status(thm)],[8])).
% 0.80/1.00 tff(10,plain,
% 0.80/1.00 (^[A: $i, B: $i] : refl(((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B)) <=> ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B)))),
% 0.80/1.00 inference(bind,[status(th)],[])).
% 0.80/1.00 tff(11,plain,
% 0.80/1.00 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B)) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 0.80/1.00 inference(quant_intro,[status(thm)],[10])).
% 0.80/1.00 tff(12,plain,
% 0.80/1.00 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B)) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 0.80/1.00 inference(rewrite,[status(thm)],[])).
% 0.80/1.00 tff(13,plain,
% 0.80/1.00 (^[A: $i, B: $i] : rewrite((element(B, powerset(powerset(A))) => (complements_of_subsets(A, complements_of_subsets(A, B)) = B)) <=> ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B)))),
% 0.80/1.00 inference(bind,[status(th)],[])).
% 0.80/1.00 tff(14,plain,
% 0.80/1.00 (![A: $i, B: $i] : (element(B, powerset(powerset(A))) => (complements_of_subsets(A, complements_of_subsets(A, B)) = B)) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 0.80/1.00 inference(quant_intro,[status(thm)],[13])).
% 0.80/1.00 tff(15,axiom,(![A: $i, B: $i] : (element(B, powerset(powerset(A))) => (complements_of_subsets(A, complements_of_subsets(A, B)) = B))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','involutiveness_k7_setfam_1')).
% 0.80/1.00 tff(16,plain,
% 0.80/1.00 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 0.80/1.00 inference(modus_ponens,[status(thm)],[15, 14])).
% 0.80/1.00 tff(17,plain,
% 0.80/1.00 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 0.80/1.00 inference(modus_ponens,[status(thm)],[16, 12])).
% 0.80/1.00 tff(18,plain,(
% 0.80/1.00 ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 0.80/1.00 inference(skolemize,[status(sab)],[17])).
% 0.80/1.00 tff(19,plain,
% 0.80/1.00 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 0.80/1.00 inference(modus_ponens,[status(thm)],[18, 11])).
% 0.80/1.00 tff(20,plain,
% 0.80/1.00 (((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))) | ((~element(B!4, powerset(powerset(A!5)))) | (complements_of_subsets(A!5, complements_of_subsets(A!5, B!4)) = B!4))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))) | (~element(B!4, powerset(powerset(A!5)))) | (complements_of_subsets(A!5, complements_of_subsets(A!5, B!4)) = B!4))),
% 0.80/1.00 inference(rewrite,[status(thm)],[])).
% 0.80/1.00 tff(21,plain,
% 0.80/1.00 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))) | ((~element(B!4, powerset(powerset(A!5)))) | (complements_of_subsets(A!5, complements_of_subsets(A!5, B!4)) = B!4))),
% 0.80/1.00 inference(quant_inst,[status(thm)],[])).
% 0.80/1.00 tff(22,plain,
% 0.80/1.00 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))) | (~element(B!4, powerset(powerset(A!5)))) | (complements_of_subsets(A!5, complements_of_subsets(A!5, B!4)) = B!4)),
% 0.80/1.00 inference(modus_ponens,[status(thm)],[21, 20])).
% 0.80/1.00 tff(23,plain,
% 0.80/1.00 (complements_of_subsets(A!5, complements_of_subsets(A!5, B!4)) = B!4),
% 0.80/1.00 inference(unit_resolution,[status(thm)],[22, 19, 9])).
% 0.80/1.00 tff(24,plain,
% 0.80/1.00 ((complements_of_subsets(A!5, complements_of_subsets(A!5, B!4)) = empty_set) <=> (B!4 = empty_set)),
% 0.80/1.00 inference(monotonicity,[status(thm)],[23])).
% 0.80/1.00 tff(25,plain,
% 0.80/1.00 ((B!4 = empty_set) <=> (complements_of_subsets(A!5, complements_of_subsets(A!5, B!4)) = empty_set)),
% 0.80/1.00 inference(symmetry,[status(thm)],[24])).
% 0.80/1.00 tff(26,assumption,(~((~(B!4 = empty_set)) | (complements_of_subsets(A!5, B!4) = empty_set))), introduced(assumption)).
% 0.80/1.00 tff(27,plain,
% 0.80/1.00 (((~(B!4 = empty_set)) | (complements_of_subsets(A!5, B!4) = empty_set)) | (B!4 = empty_set)),
% 0.80/1.00 inference(tautology,[status(thm)],[])).
% 0.80/1.00 tff(28,plain,
% 0.80/1.00 (B!4 = empty_set),
% 0.80/1.00 inference(unit_resolution,[status(thm)],[27, 26])).
% 0.80/1.00 tff(29,plain,
% 0.80/1.00 (complements_of_subsets(A!5, complements_of_subsets(A!5, B!4)) = empty_set),
% 0.80/1.00 inference(modus_ponens,[status(thm)],[28, 25])).
% 0.80/1.00 tff(30,plain,
% 0.80/1.00 (((~(B!4 = empty_set)) | (complements_of_subsets(A!5, B!4) = empty_set)) | (~(complements_of_subsets(A!5, B!4) = empty_set))),
% 0.80/1.00 inference(tautology,[status(thm)],[])).
% 0.80/1.00 tff(31,plain,
% 0.80/1.00 (~(complements_of_subsets(A!5, B!4) = empty_set)),
% 0.80/1.00 inference(unit_resolution,[status(thm)],[30, 26])).
% 0.80/1.00 tff(32,plain,
% 0.80/1.00 (^[A: $i, B: $i] : refl(((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A)))) <=> ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A)))))),
% 0.87/1.01 inference(bind,[status(th)],[])).
% 0.87/1.01 tff(33,plain,
% 0.87/1.01 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A)))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 0.87/1.01 inference(quant_intro,[status(thm)],[32])).
% 0.87/1.01 tff(34,plain,
% 0.87/1.01 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A)))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 0.87/1.01 inference(rewrite,[status(thm)],[])).
% 0.87/1.01 tff(35,plain,
% 0.87/1.01 (^[A: $i, B: $i] : rewrite((element(B, powerset(powerset(A))) => element(complements_of_subsets(A, B), powerset(powerset(A)))) <=> ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A)))))),
% 0.87/1.01 inference(bind,[status(th)],[])).
% 0.87/1.01 tff(36,plain,
% 0.87/1.01 (![A: $i, B: $i] : (element(B, powerset(powerset(A))) => element(complements_of_subsets(A, B), powerset(powerset(A)))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 0.87/1.01 inference(quant_intro,[status(thm)],[35])).
% 0.87/1.01 tff(37,axiom,(![A: $i, B: $i] : (element(B, powerset(powerset(A))) => element(complements_of_subsets(A, B), powerset(powerset(A))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','dt_k7_setfam_1')).
% 0.87/1.01 tff(38,plain,
% 0.87/1.01 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 0.87/1.01 inference(modus_ponens,[status(thm)],[37, 36])).
% 0.87/1.01 tff(39,plain,
% 0.87/1.01 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 0.87/1.01 inference(modus_ponens,[status(thm)],[38, 34])).
% 0.87/1.01 tff(40,plain,(
% 0.87/1.01 ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 0.87/1.01 inference(skolemize,[status(sab)],[39])).
% 0.87/1.01 tff(41,plain,
% 0.87/1.01 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 0.87/1.01 inference(modus_ponens,[status(thm)],[40, 33])).
% 0.87/1.01 tff(42,plain,
% 0.87/1.01 (((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))) | ((~element(B!4, powerset(powerset(A!5)))) | element(complements_of_subsets(A!5, B!4), powerset(powerset(A!5))))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))) | (~element(B!4, powerset(powerset(A!5)))) | element(complements_of_subsets(A!5, B!4), powerset(powerset(A!5))))),
% 0.87/1.01 inference(rewrite,[status(thm)],[])).
% 0.87/1.01 tff(43,plain,
% 0.87/1.01 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))) | ((~element(B!4, powerset(powerset(A!5)))) | element(complements_of_subsets(A!5, B!4), powerset(powerset(A!5))))),
% 0.87/1.01 inference(quant_inst,[status(thm)],[])).
% 0.87/1.01 tff(44,plain,
% 0.87/1.01 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))) | (~element(B!4, powerset(powerset(A!5)))) | element(complements_of_subsets(A!5, B!4), powerset(powerset(A!5)))),
% 0.87/1.01 inference(modus_ponens,[status(thm)],[43, 42])).
% 0.87/1.01 tff(45,plain,
% 0.87/1.01 (element(complements_of_subsets(A!5, B!4), powerset(powerset(A!5)))),
% 0.87/1.01 inference(unit_resolution,[status(thm)],[44, 41, 9])).
% 0.87/1.01 tff(46,plain,
% 0.87/1.01 (^[A: $i, B: $i] : refl(((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A))))) <=> ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A))))))),
% 0.87/1.01 inference(bind,[status(th)],[])).
% 0.87/1.01 tff(47,plain,
% 0.87/1.01 (![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A))))) <=> ![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))),
% 0.87/1.01 inference(quant_intro,[status(thm)],[46])).
% 0.87/1.01 tff(48,plain,
% 0.87/1.01 (^[A: $i, B: $i] : trans(monotonicity(trans(monotonicity(rewrite(((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set)) <=> (~((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set))))), ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) <=> (~(~((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set))))))), rewrite((~(~((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set))))) <=> ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)))), ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) <=> ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set))))), (((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A))))) <=> (((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A))))))), rewrite((((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A))))) <=> ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))), (((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A))))) <=> ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))))),
% 0.87/1.01 inference(bind,[status(th)],[])).
% 0.87/1.01 tff(49,plain,
% 0.87/1.01 (![A: $i, B: $i] : ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A))))) <=> ![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))),
% 0.87/1.01 inference(quant_intro,[status(thm)],[48])).
% 0.87/1.01 tff(50,plain,
% 0.87/1.01 (![A: $i, B: $i] : ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A))))) <=> ![A: $i, B: $i] : ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A)))))),
% 0.87/1.01 inference(rewrite,[status(thm)],[])).
% 0.87/1.01 tff(51,plain,
% 0.87/1.01 (^[A: $i, B: $i] : rewrite((element(B, powerset(powerset(A))) => (~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set)))) <=> ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A))))))),
% 0.87/1.01 inference(bind,[status(th)],[])).
% 0.87/1.01 tff(52,plain,
% 0.87/1.01 (![A: $i, B: $i] : (element(B, powerset(powerset(A))) => (~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set)))) <=> ![A: $i, B: $i] : ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A)))))),
% 0.87/1.01 inference(quant_intro,[status(thm)],[51])).
% 0.87/1.01 tff(53,axiom,(![A: $i, B: $i] : (element(B, powerset(powerset(A))) => (~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t46_setfam_1')).
% 0.87/1.01 tff(54,plain,
% 0.87/1.01 (![A: $i, B: $i] : ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A)))))),
% 0.87/1.01 inference(modus_ponens,[status(thm)],[53, 52])).
% 0.87/1.01 tff(55,plain,
% 0.87/1.01 (![A: $i, B: $i] : ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A)))))),
% 0.87/1.01 inference(modus_ponens,[status(thm)],[54, 50])).
% 0.87/1.01 tff(56,plain,(
% 0.87/1.01 ![A: $i, B: $i] : ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A)))))),
% 0.87/1.01 inference(skolemize,[status(sab)],[55])).
% 0.87/1.01 tff(57,plain,
% 0.87/1.01 (![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))),
% 0.87/1.01 inference(modus_ponens,[status(thm)],[56, 49])).
% 0.87/1.01 tff(58,plain,
% 0.87/1.01 (![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))),
% 0.87/1.01 inference(modus_ponens,[status(thm)],[57, 47])).
% 0.87/1.01 tff(59,plain,
% 0.87/1.01 (((~![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))) | ((complements_of_subsets(A!5, B!4) = empty_set) | (~(complements_of_subsets(A!5, complements_of_subsets(A!5, B!4)) = empty_set)) | (~element(complements_of_subsets(A!5, B!4), powerset(powerset(A!5)))))) <=> ((~![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))) | (complements_of_subsets(A!5, B!4) = empty_set) | (~(complements_of_subsets(A!5, complements_of_subsets(A!5, B!4)) = empty_set)) | (~element(complements_of_subsets(A!5, B!4), powerset(powerset(A!5)))))),
% 0.87/1.01 inference(rewrite,[status(thm)],[])).
% 0.87/1.01 tff(60,plain,
% 0.87/1.01 ((~![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))) | ((complements_of_subsets(A!5, B!4) = empty_set) | (~(complements_of_subsets(A!5, complements_of_subsets(A!5, B!4)) = empty_set)) | (~element(complements_of_subsets(A!5, B!4), powerset(powerset(A!5)))))),
% 0.87/1.01 inference(quant_inst,[status(thm)],[])).
% 0.87/1.01 tff(61,plain,
% 0.87/1.01 ((~![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))) | (complements_of_subsets(A!5, B!4) = empty_set) | (~(complements_of_subsets(A!5, complements_of_subsets(A!5, B!4)) = empty_set)) | (~element(complements_of_subsets(A!5, B!4), powerset(powerset(A!5))))),
% 0.87/1.01 inference(modus_ponens,[status(thm)],[60, 59])).
% 0.87/1.01 tff(62,plain,
% 0.87/1.01 ((complements_of_subsets(A!5, B!4) = empty_set) | (~(complements_of_subsets(A!5, complements_of_subsets(A!5, B!4)) = empty_set)) | (~element(complements_of_subsets(A!5, B!4), powerset(powerset(A!5))))),
% 0.87/1.01 inference(unit_resolution,[status(thm)],[61, 58])).
% 0.87/1.01 tff(63,plain,
% 0.87/1.01 ((complements_of_subsets(A!5, B!4) = empty_set) | (~(complements_of_subsets(A!5, complements_of_subsets(A!5, B!4)) = empty_set))),
% 0.87/1.01 inference(unit_resolution,[status(thm)],[62, 45])).
% 0.87/1.01 tff(64,plain,
% 0.87/1.01 (~(complements_of_subsets(A!5, complements_of_subsets(A!5, B!4)) = empty_set)),
% 0.87/1.01 inference(unit_resolution,[status(thm)],[63, 31])).
% 0.87/1.01 tff(65,plain,
% 0.87/1.01 ($false),
% 0.87/1.01 inference(unit_resolution,[status(thm)],[64, 29])).
% 0.87/1.01 tff(66,plain,((~(B!4 = empty_set)) | (complements_of_subsets(A!5, B!4) = empty_set)), inference(lemma,lemma(discharge,[]))).
% 0.87/1.01 tff(67,plain,
% 0.87/1.01 ((~(~((~((B!4 = empty_set) | (~(complements_of_subsets(A!5, B!4) = empty_set)))) | (~((~(B!4 = empty_set)) | (complements_of_subsets(A!5, B!4) = empty_set)))))) <=> ((~((B!4 = empty_set) | (~(complements_of_subsets(A!5, B!4) = empty_set)))) | (~((~(B!4 = empty_set)) | (complements_of_subsets(A!5, B!4) = empty_set))))),
% 0.87/1.01 inference(rewrite,[status(thm)],[])).
% 0.87/1.01 tff(68,plain,
% 0.87/1.01 ((((B!4 = empty_set) | (~(complements_of_subsets(A!5, B!4) = empty_set))) & ((~(B!4 = empty_set)) | (complements_of_subsets(A!5, B!4) = empty_set))) <=> (~((~((B!4 = empty_set) | (~(complements_of_subsets(A!5, B!4) = empty_set)))) | (~((~(B!4 = empty_set)) | (complements_of_subsets(A!5, B!4) = empty_set)))))),
% 0.87/1.01 inference(rewrite,[status(thm)],[])).
% 0.87/1.01 tff(69,plain,
% 0.87/1.01 ((~(~((~(B!4 = empty_set)) | (complements_of_subsets(A!5, B!4) = empty_set)))) <=> ((~(B!4 = empty_set)) | (complements_of_subsets(A!5, B!4) = empty_set))),
% 0.87/1.01 inference(rewrite,[status(thm)],[])).
% 0.87/1.01 tff(70,plain,
% 0.87/1.01 (((~(complements_of_subsets(A!5, B!4) = empty_set)) & (B!4 = empty_set)) <=> (~((~(B!4 = empty_set)) | (complements_of_subsets(A!5, B!4) = empty_set)))),
% 0.87/1.01 inference(rewrite,[status(thm)],[])).
% 0.87/1.01 tff(71,plain,
% 0.87/1.01 ((~((~(complements_of_subsets(A!5, B!4) = empty_set)) & (B!4 = empty_set))) <=> (~(~((~(B!4 = empty_set)) | (complements_of_subsets(A!5, B!4) = empty_set))))),
% 0.87/1.01 inference(monotonicity,[status(thm)],[70])).
% 0.87/1.01 tff(72,plain,
% 0.87/1.01 ((~((~(complements_of_subsets(A!5, B!4) = empty_set)) & (B!4 = empty_set))) <=> ((~(B!4 = empty_set)) | (complements_of_subsets(A!5, B!4) = empty_set))),
% 0.87/1.01 inference(transitivity,[status(thm)],[71, 69])).
% 0.87/1.01 tff(73,plain,
% 0.87/1.01 ((~(~((B!4 = empty_set) | (~(complements_of_subsets(A!5, B!4) = empty_set))))) <=> ((B!4 = empty_set) | (~(complements_of_subsets(A!5, B!4) = empty_set)))),
% 0.87/1.01 inference(rewrite,[status(thm)],[])).
% 0.87/1.01 tff(74,plain,
% 0.87/1.01 (((~(B!4 = empty_set)) & (complements_of_subsets(A!5, B!4) = empty_set)) <=> (~((B!4 = empty_set) | (~(complements_of_subsets(A!5, B!4) = empty_set))))),
% 0.87/1.01 inference(rewrite,[status(thm)],[])).
% 0.87/1.01 tff(75,plain,
% 0.87/1.01 ((~((~(B!4 = empty_set)) & (complements_of_subsets(A!5, B!4) = empty_set))) <=> (~(~((B!4 = empty_set) | (~(complements_of_subsets(A!5, B!4) = empty_set)))))),
% 0.87/1.01 inference(monotonicity,[status(thm)],[74])).
% 0.87/1.01 tff(76,plain,
% 0.87/1.01 ((~((~(B!4 = empty_set)) & (complements_of_subsets(A!5, B!4) = empty_set))) <=> ((B!4 = empty_set) | (~(complements_of_subsets(A!5, B!4) = empty_set)))),
% 0.87/1.01 inference(transitivity,[status(thm)],[75, 73])).
% 0.87/1.01 tff(77,plain,
% 0.87/1.01 (((~((~(B!4 = empty_set)) & (complements_of_subsets(A!5, B!4) = empty_set))) & (~((~(complements_of_subsets(A!5, B!4) = empty_set)) & (B!4 = empty_set)))) <=> (((B!4 = empty_set) | (~(complements_of_subsets(A!5, B!4) = empty_set))) & ((~(B!4 = empty_set)) | (complements_of_subsets(A!5, B!4) = empty_set)))),
% 0.87/1.01 inference(monotonicity,[status(thm)],[76, 72])).
% 0.87/1.01 tff(78,plain,
% 0.87/1.01 (((~((~(B!4 = empty_set)) & (complements_of_subsets(A!5, B!4) = empty_set))) & (~((~(complements_of_subsets(A!5, B!4) = empty_set)) & (B!4 = empty_set)))) <=> (~((~((B!4 = empty_set) | (~(complements_of_subsets(A!5, B!4) = empty_set)))) | (~((~(B!4 = empty_set)) | (complements_of_subsets(A!5, B!4) = empty_set)))))),
% 0.87/1.01 inference(transitivity,[status(thm)],[77, 68])).
% 0.87/1.01 tff(79,plain,
% 0.87/1.01 ((~((~((~(B!4 = empty_set)) & (complements_of_subsets(A!5, B!4) = empty_set))) & (~((~(complements_of_subsets(A!5, B!4) = empty_set)) & (B!4 = empty_set))))) <=> (~(~((~((B!4 = empty_set) | (~(complements_of_subsets(A!5, B!4) = empty_set)))) | (~((~(B!4 = empty_set)) | (complements_of_subsets(A!5, B!4) = empty_set))))))),
% 0.87/1.01 inference(monotonicity,[status(thm)],[78])).
% 0.87/1.01 tff(80,plain,
% 0.87/1.01 ((~((~((~(B!4 = empty_set)) & (complements_of_subsets(A!5, B!4) = empty_set))) & (~((~(complements_of_subsets(A!5, B!4) = empty_set)) & (B!4 = empty_set))))) <=> ((~((B!4 = empty_set) | (~(complements_of_subsets(A!5, B!4) = empty_set)))) | (~((~(B!4 = empty_set)) | (complements_of_subsets(A!5, B!4) = empty_set))))),
% 0.87/1.01 inference(transitivity,[status(thm)],[79, 67])).
% 0.87/1.01 tff(81,plain,
% 0.87/1.01 (~((~((~(B!4 = empty_set)) & (complements_of_subsets(A!5, B!4) = empty_set))) & (~((~(complements_of_subsets(A!5, B!4) = empty_set)) & (B!4 = empty_set))))),
% 0.87/1.01 inference(or_elim,[status(thm)],[8])).
% 0.87/1.01 tff(82,plain,
% 0.87/1.01 ((~((B!4 = empty_set) | (~(complements_of_subsets(A!5, B!4) = empty_set)))) | (~((~(B!4 = empty_set)) | (complements_of_subsets(A!5, B!4) = empty_set)))),
% 0.87/1.01 inference(modus_ponens,[status(thm)],[81, 80])).
% 0.87/1.01 tff(83,plain,
% 0.87/1.01 (~((B!4 = empty_set) | (~(complements_of_subsets(A!5, B!4) = empty_set)))),
% 0.87/1.01 inference(unit_resolution,[status(thm)],[82, 66])).
% 0.87/1.01 tff(84,plain,
% 0.87/1.01 (((B!4 = empty_set) | (~(complements_of_subsets(A!5, B!4) = empty_set))) | (~(B!4 = empty_set))),
% 0.87/1.01 inference(tautology,[status(thm)],[])).
% 0.87/1.01 tff(85,plain,
% 0.87/1.01 (~(B!4 = empty_set)),
% 0.87/1.01 inference(unit_resolution,[status(thm)],[84, 83])).
% 0.87/1.01 tff(86,plain,
% 0.87/1.01 (((B!4 = empty_set) | (~(complements_of_subsets(A!5, B!4) = empty_set))) | (complements_of_subsets(A!5, B!4) = empty_set)),
% 0.87/1.01 inference(tautology,[status(thm)],[])).
% 0.87/1.01 tff(87,plain,
% 0.87/1.01 (complements_of_subsets(A!5, B!4) = empty_set),
% 0.87/1.01 inference(unit_resolution,[status(thm)],[86, 83])).
% 0.87/1.01 tff(88,plain,
% 0.87/1.01 (((~![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))) | ((~element(B!4, powerset(powerset(A!5)))) | (B!4 = empty_set) | (~(complements_of_subsets(A!5, B!4) = empty_set)))) <=> ((~![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))) | (~element(B!4, powerset(powerset(A!5)))) | (B!4 = empty_set) | (~(complements_of_subsets(A!5, B!4) = empty_set)))),
% 0.87/1.01 inference(rewrite,[status(thm)],[])).
% 0.87/1.01 tff(89,plain,
% 0.87/1.01 (((B!4 = empty_set) | (~(complements_of_subsets(A!5, B!4) = empty_set)) | (~element(B!4, powerset(powerset(A!5))))) <=> ((~element(B!4, powerset(powerset(A!5)))) | (B!4 = empty_set) | (~(complements_of_subsets(A!5, B!4) = empty_set)))),
% 0.87/1.01 inference(rewrite,[status(thm)],[])).
% 0.87/1.01 tff(90,plain,
% 0.87/1.01 (((~![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))) | ((B!4 = empty_set) | (~(complements_of_subsets(A!5, B!4) = empty_set)) | (~element(B!4, powerset(powerset(A!5)))))) <=> ((~![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))) | ((~element(B!4, powerset(powerset(A!5)))) | (B!4 = empty_set) | (~(complements_of_subsets(A!5, B!4) = empty_set))))),
% 0.87/1.01 inference(monotonicity,[status(thm)],[89])).
% 0.87/1.01 tff(91,plain,
% 0.87/1.01 (((~![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))) | ((B!4 = empty_set) | (~(complements_of_subsets(A!5, B!4) = empty_set)) | (~element(B!4, powerset(powerset(A!5)))))) <=> ((~![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))) | (~element(B!4, powerset(powerset(A!5)))) | (B!4 = empty_set) | (~(complements_of_subsets(A!5, B!4) = empty_set)))),
% 0.87/1.01 inference(transitivity,[status(thm)],[90, 88])).
% 0.87/1.01 tff(92,plain,
% 0.87/1.01 ((~![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))) | ((B!4 = empty_set) | (~(complements_of_subsets(A!5, B!4) = empty_set)) | (~element(B!4, powerset(powerset(A!5)))))),
% 0.87/1.01 inference(quant_inst,[status(thm)],[])).
% 0.87/1.01 tff(93,plain,
% 0.87/1.01 ((~![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))) | (~element(B!4, powerset(powerset(A!5)))) | (B!4 = empty_set) | (~(complements_of_subsets(A!5, B!4) = empty_set))),
% 0.87/1.01 inference(modus_ponens,[status(thm)],[92, 91])).
% 0.87/1.01 tff(94,plain,
% 0.87/1.01 ((B!4 = empty_set) | (~(complements_of_subsets(A!5, B!4) = empty_set))),
% 0.87/1.01 inference(unit_resolution,[status(thm)],[93, 9, 58])).
% 0.87/1.01 tff(95,plain,
% 0.87/1.01 ($false),
% 0.87/1.01 inference(unit_resolution,[status(thm)],[94, 87, 85])).
% 0.87/1.01 % SZS output end Proof
% 0.87/1.02 % E exiting
%------------------------------------------------------------------------------