%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : NUM409+1 : TPTP v9.0.0. Released v3.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n013.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 05:20:54 AM UTC 2025
% Result : Theorem 0.23s 0.42s
% Output : Proof 0.23s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.14 % Problem : NUM409+1 : TPTP v9.0.0. Released v3.2.0.
% 0.04/0.14 % Command : run_E %s %d THM
% 0.15/0.36 % Computer : n013.cluster.edu
% 0.15/0.36 % Model : x86_64 x86_64
% 0.15/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.36 % Memory : 8042.1875MB
% 0.15/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.36 % CPULimit : 300
% 0.15/0.36 % WCLimit : 300
% 0.15/0.36 % DateTime : Fri Jun 20 01:02:40 EDT 2025
% 0.15/0.36 % CPUTime :
% 0.23/0.42 % SZS status Theorem
% 0.23/0.42 % SZS output start Proof
% 0.23/0.42 tff(subset_type, type, (
% 0.23/0.42 subset: ( $i * $i ) > $o)).
% 0.23/0.42 tff(tptp_fun_A_19_type, type, (
% 0.23/0.42 tptp_fun_A_19: $i)).
% 0.23/0.42 tff(empty_set_type, type, (
% 0.23/0.42 empty_set: $i)).
% 0.23/0.42 tff(relation_rng_type, type, (
% 0.23/0.42 relation_rng: $i > $i)).
% 0.23/0.42 tff(relation_dom_type, type, (
% 0.23/0.42 relation_dom: $i > $i)).
% 0.23/0.42 tff(empty_type, type, (
% 0.23/0.42 empty: $i > $o)).
% 0.23/0.42 tff(relation_type, type, (
% 0.23/0.42 relation: $i > $o)).
% 0.23/0.42 tff(relation_empty_yielding_type, type, (
% 0.23/0.42 relation_empty_yielding: $i > $o)).
% 0.23/0.42 tff(transfinite_sequence_of_type, type, (
% 0.23/0.42 transfinite_sequence_of: ( $i * $i ) > $o)).
% 0.23/0.42 tff(transfinite_sequence_type, type, (
% 0.23/0.42 transfinite_sequence: $i > $o)).
% 0.23/0.42 tff(ordinal_type, type, (
% 0.23/0.42 ordinal: $i > $o)).
% 0.23/0.42 tff(function_type, type, (
% 0.23/0.42 function: $i > $o)).
% 0.23/0.42 tff(one_to_one_type, type, (
% 0.23/0.42 one_to_one: $i > $o)).
% 0.23/0.42 tff(epsilon_transitive_type, type, (
% 0.23/0.42 epsilon_transitive: $i > $o)).
% 0.23/0.42 tff(epsilon_connected_type, type, (
% 0.23/0.42 epsilon_connected: $i > $o)).
% 0.23/0.42 tff(1,plain,
% 0.23/0.42 (^[A: $i] : refl(((~empty(A)) | (~((~empty(relation_rng(A))) | (~relation(relation_rng(A)))))) <=> ((~empty(A)) | (~((~empty(relation_rng(A))) | (~relation(relation_rng(A)))))))),
% 0.23/0.42 inference(bind,[status(th)],[])).
% 0.23/0.42 tff(2,plain,
% 0.23/0.42 (![A: $i] : ((~empty(A)) | (~((~empty(relation_rng(A))) | (~relation(relation_rng(A)))))) <=> ![A: $i] : ((~empty(A)) | (~((~empty(relation_rng(A))) | (~relation(relation_rng(A))))))),
% 0.23/0.42 inference(quant_intro,[status(thm)],[1])).
% 0.23/0.42 tff(3,plain,
% 0.23/0.42 (^[A: $i] : rewrite(((~empty(A)) | (empty(relation_rng(A)) & relation(relation_rng(A)))) <=> ((~empty(A)) | (~((~empty(relation_rng(A))) | (~relation(relation_rng(A)))))))),
% 0.23/0.42 inference(bind,[status(th)],[])).
% 0.23/0.42 tff(4,plain,
% 0.23/0.42 (![A: $i] : ((~empty(A)) | (empty(relation_rng(A)) & relation(relation_rng(A)))) <=> ![A: $i] : ((~empty(A)) | (~((~empty(relation_rng(A))) | (~relation(relation_rng(A))))))),
% 0.23/0.42 inference(quant_intro,[status(thm)],[3])).
% 0.23/0.42 tff(5,plain,
% 0.23/0.42 (![A: $i] : ((~empty(A)) | (empty(relation_rng(A)) & relation(relation_rng(A)))) <=> ![A: $i] : ((~empty(A)) | (empty(relation_rng(A)) & relation(relation_rng(A))))),
% 0.23/0.42 inference(rewrite,[status(thm)],[])).
% 0.23/0.42 tff(6,plain,
% 0.23/0.42 (^[A: $i] : rewrite((empty(A) => (empty(relation_rng(A)) & relation(relation_rng(A)))) <=> ((~empty(A)) | (empty(relation_rng(A)) & relation(relation_rng(A)))))),
% 0.23/0.42 inference(bind,[status(th)],[])).
% 0.23/0.42 tff(7,plain,
% 0.23/0.42 (![A: $i] : (empty(A) => (empty(relation_rng(A)) & relation(relation_rng(A)))) <=> ![A: $i] : ((~empty(A)) | (empty(relation_rng(A)) & relation(relation_rng(A))))),
% 0.23/0.42 inference(quant_intro,[status(thm)],[6])).
% 0.23/0.42 tff(8,axiom,(![A: $i] : (empty(A) => (empty(relation_rng(A)) & relation(relation_rng(A))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','fc8_relat_1')).
% 0.23/0.42 tff(9,plain,
% 0.23/0.42 (![A: $i] : ((~empty(A)) | (empty(relation_rng(A)) & relation(relation_rng(A))))),
% 0.23/0.42 inference(modus_ponens,[status(thm)],[8, 7])).
% 0.23/0.42 tff(10,plain,
% 0.23/0.42 (![A: $i] : ((~empty(A)) | (empty(relation_rng(A)) & relation(relation_rng(A))))),
% 0.23/0.42 inference(modus_ponens,[status(thm)],[9, 5])).
% 0.23/0.42 tff(11,plain,(
% 0.23/0.42 ![A: $i] : ((~empty(A)) | (empty(relation_rng(A)) & relation(relation_rng(A))))),
% 0.23/0.42 inference(skolemize,[status(sab)],[10])).
% 0.23/0.42 tff(12,plain,
% 0.23/0.42 (![A: $i] : ((~empty(A)) | (~((~empty(relation_rng(A))) | (~relation(relation_rng(A))))))),
% 0.23/0.42 inference(modus_ponens,[status(thm)],[11, 4])).
% 0.23/0.42 tff(13,plain,
% 0.23/0.42 (![A: $i] : ((~empty(A)) | (~((~empty(relation_rng(A))) | (~relation(relation_rng(A))))))),
% 0.23/0.42 inference(modus_ponens,[status(thm)],[12, 2])).
% 0.23/0.42 tff(14,plain,
% 0.23/0.42 (empty(empty_set) <=> empty(empty_set)),
% 0.23/0.42 inference(rewrite,[status(thm)],[])).
% 0.23/0.42 tff(15,axiom,((empty(empty_set) & relation(empty_set)) & relation_empty_yielding(empty_set)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','fc12_relat_1')).
% 0.23/0.42 tff(16,plain,
% 0.23/0.42 (empty(empty_set) & relation(empty_set)),
% 0.23/0.42 inference(and_elim,[status(thm)],[15])).
% 0.23/0.42 tff(17,plain,
% 0.23/0.42 (empty(empty_set)),
% 0.23/0.42 inference(and_elim,[status(thm)],[16])).
% 0.23/0.42 tff(18,plain,
% 0.23/0.42 (empty(empty_set)),
% 0.23/0.42 inference(modus_ponens,[status(thm)],[17, 14])).
% 0.23/0.42 tff(19,plain,
% 0.23/0.42 (((~![A: $i] : ((~empty(A)) | (~((~empty(relation_rng(A))) | (~relation(relation_rng(A))))))) | ((~empty(empty_set)) | (~((~empty(relation_rng(empty_set))) | (~relation(relation_rng(empty_set))))))) <=> ((~![A: $i] : ((~empty(A)) | (~((~empty(relation_rng(A))) | (~relation(relation_rng(A))))))) | (~empty(empty_set)) | (~((~empty(relation_rng(empty_set))) | (~relation(relation_rng(empty_set))))))),
% 0.23/0.42 inference(rewrite,[status(thm)],[])).
% 0.23/0.42 tff(20,plain,
% 0.23/0.42 ((~![A: $i] : ((~empty(A)) | (~((~empty(relation_rng(A))) | (~relation(relation_rng(A))))))) | ((~empty(empty_set)) | (~((~empty(relation_rng(empty_set))) | (~relation(relation_rng(empty_set))))))),
% 0.23/0.42 inference(quant_inst,[status(thm)],[])).
% 0.23/0.42 tff(21,plain,
% 0.23/0.42 ((~![A: $i] : ((~empty(A)) | (~((~empty(relation_rng(A))) | (~relation(relation_rng(A))))))) | (~empty(empty_set)) | (~((~empty(relation_rng(empty_set))) | (~relation(relation_rng(empty_set)))))),
% 0.23/0.42 inference(modus_ponens,[status(thm)],[20, 19])).
% 0.23/0.42 tff(22,plain,
% 0.23/0.42 (~((~empty(relation_rng(empty_set))) | (~relation(relation_rng(empty_set))))),
% 0.23/0.42 inference(unit_resolution,[status(thm)],[21, 18, 13])).
% 0.23/0.42 tff(23,plain,
% 0.23/0.42 (((~empty(relation_rng(empty_set))) | (~relation(relation_rng(empty_set)))) | empty(relation_rng(empty_set))),
% 0.23/0.42 inference(tautology,[status(thm)],[])).
% 0.23/0.42 tff(24,plain,
% 0.23/0.42 (empty(relation_rng(empty_set))),
% 0.23/0.42 inference(unit_resolution,[status(thm)],[23, 22])).
% 0.23/0.42 tff(25,plain,
% 0.23/0.42 (^[A: $i] : refl(((~empty(A)) | (~((~empty(relation_dom(A))) | (~relation(relation_dom(A)))))) <=> ((~empty(A)) | (~((~empty(relation_dom(A))) | (~relation(relation_dom(A)))))))),
% 0.23/0.42 inference(bind,[status(th)],[])).
% 0.23/0.42 tff(26,plain,
% 0.23/0.42 (![A: $i] : ((~empty(A)) | (~((~empty(relation_dom(A))) | (~relation(relation_dom(A)))))) <=> ![A: $i] : ((~empty(A)) | (~((~empty(relation_dom(A))) | (~relation(relation_dom(A))))))),
% 0.23/0.42 inference(quant_intro,[status(thm)],[25])).
% 0.23/0.42 tff(27,plain,
% 0.23/0.42 (^[A: $i] : rewrite(((~empty(A)) | (empty(relation_dom(A)) & relation(relation_dom(A)))) <=> ((~empty(A)) | (~((~empty(relation_dom(A))) | (~relation(relation_dom(A)))))))),
% 0.23/0.42 inference(bind,[status(th)],[])).
% 0.23/0.42 tff(28,plain,
% 0.23/0.42 (![A: $i] : ((~empty(A)) | (empty(relation_dom(A)) & relation(relation_dom(A)))) <=> ![A: $i] : ((~empty(A)) | (~((~empty(relation_dom(A))) | (~relation(relation_dom(A))))))),
% 0.23/0.42 inference(quant_intro,[status(thm)],[27])).
% 0.23/0.42 tff(29,plain,
% 0.23/0.42 (![A: $i] : ((~empty(A)) | (empty(relation_dom(A)) & relation(relation_dom(A)))) <=> ![A: $i] : ((~empty(A)) | (empty(relation_dom(A)) & relation(relation_dom(A))))),
% 0.23/0.42 inference(rewrite,[status(thm)],[])).
% 0.23/0.42 tff(30,plain,
% 0.23/0.42 (^[A: $i] : rewrite((empty(A) => (empty(relation_dom(A)) & relation(relation_dom(A)))) <=> ((~empty(A)) | (empty(relation_dom(A)) & relation(relation_dom(A)))))),
% 0.23/0.42 inference(bind,[status(th)],[])).
% 0.23/0.42 tff(31,plain,
% 0.23/0.42 (![A: $i] : (empty(A) => (empty(relation_dom(A)) & relation(relation_dom(A)))) <=> ![A: $i] : ((~empty(A)) | (empty(relation_dom(A)) & relation(relation_dom(A))))),
% 0.23/0.42 inference(quant_intro,[status(thm)],[30])).
% 0.23/0.42 tff(32,axiom,(![A: $i] : (empty(A) => (empty(relation_dom(A)) & relation(relation_dom(A))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','fc7_relat_1')).
% 0.23/0.42 tff(33,plain,
% 0.23/0.42 (![A: $i] : ((~empty(A)) | (empty(relation_dom(A)) & relation(relation_dom(A))))),
% 0.23/0.42 inference(modus_ponens,[status(thm)],[32, 31])).
% 0.23/0.42 tff(34,plain,
% 0.23/0.42 (![A: $i] : ((~empty(A)) | (empty(relation_dom(A)) & relation(relation_dom(A))))),
% 0.23/0.42 inference(modus_ponens,[status(thm)],[33, 29])).
% 0.23/0.42 tff(35,plain,(
% 0.23/0.42 ![A: $i] : ((~empty(A)) | (empty(relation_dom(A)) & relation(relation_dom(A))))),
% 0.23/0.42 inference(skolemize,[status(sab)],[34])).
% 0.23/0.42 tff(36,plain,
% 0.23/0.42 (![A: $i] : ((~empty(A)) | (~((~empty(relation_dom(A))) | (~relation(relation_dom(A))))))),
% 0.23/0.42 inference(modus_ponens,[status(thm)],[35, 28])).
% 0.23/0.42 tff(37,plain,
% 0.23/0.42 (![A: $i] : ((~empty(A)) | (~((~empty(relation_dom(A))) | (~relation(relation_dom(A))))))),
% 0.23/0.42 inference(modus_ponens,[status(thm)],[36, 26])).
% 0.23/0.42 tff(38,plain,
% 0.23/0.42 (((~![A: $i] : ((~empty(A)) | (~((~empty(relation_dom(A))) | (~relation(relation_dom(A))))))) | ((~empty(empty_set)) | (~((~empty(relation_dom(empty_set))) | (~relation(relation_dom(empty_set))))))) <=> ((~![A: $i] : ((~empty(A)) | (~((~empty(relation_dom(A))) | (~relation(relation_dom(A))))))) | (~empty(empty_set)) | (~((~empty(relation_dom(empty_set))) | (~relation(relation_dom(empty_set))))))),
% 0.23/0.43 inference(rewrite,[status(thm)],[])).
% 0.23/0.43 tff(39,plain,
% 0.23/0.43 ((~![A: $i] : ((~empty(A)) | (~((~empty(relation_dom(A))) | (~relation(relation_dom(A))))))) | ((~empty(empty_set)) | (~((~empty(relation_dom(empty_set))) | (~relation(relation_dom(empty_set))))))),
% 0.23/0.43 inference(quant_inst,[status(thm)],[])).
% 0.23/0.43 tff(40,plain,
% 0.23/0.43 ((~![A: $i] : ((~empty(A)) | (~((~empty(relation_dom(A))) | (~relation(relation_dom(A))))))) | (~empty(empty_set)) | (~((~empty(relation_dom(empty_set))) | (~relation(relation_dom(empty_set)))))),
% 0.23/0.43 inference(modus_ponens,[status(thm)],[39, 38])).
% 0.23/0.43 tff(41,plain,
% 0.23/0.43 (~((~empty(relation_dom(empty_set))) | (~relation(relation_dom(empty_set))))),
% 0.23/0.43 inference(unit_resolution,[status(thm)],[40, 18, 37])).
% 0.23/0.43 tff(42,plain,
% 0.23/0.43 (((~empty(relation_dom(empty_set))) | (~relation(relation_dom(empty_set)))) | empty(relation_dom(empty_set))),
% 0.23/0.43 inference(tautology,[status(thm)],[])).
% 0.23/0.43 tff(43,plain,
% 0.23/0.43 (empty(relation_dom(empty_set))),
% 0.23/0.43 inference(unit_resolution,[status(thm)],[42, 41])).
% 0.23/0.43 tff(44,plain,
% 0.23/0.43 (^[A: $i, B: $i] : refl(((~empty(B)) | (A = B) | (~empty(A))) <=> ((~empty(B)) | (A = B) | (~empty(A))))),
% 0.23/0.43 inference(bind,[status(th)],[])).
% 0.23/0.43 tff(45,plain,
% 0.23/0.43 (![A: $i, B: $i] : ((~empty(B)) | (A = B) | (~empty(A))) <=> ![A: $i, B: $i] : ((~empty(B)) | (A = B) | (~empty(A)))),
% 0.23/0.43 inference(quant_intro,[status(thm)],[44])).
% 0.23/0.43 tff(46,plain,
% 0.23/0.43 (^[A: $i, B: $i] : trans(monotonicity(rewrite((empty(A) & (~(A = B)) & empty(B)) <=> (~((~empty(B)) | (A = B) | (~empty(A))))), ((~(empty(A) & (~(A = B)) & empty(B))) <=> (~(~((~empty(B)) | (A = B) | (~empty(A))))))), rewrite((~(~((~empty(B)) | (A = B) | (~empty(A))))) <=> ((~empty(B)) | (A = B) | (~empty(A)))), ((~(empty(A) & (~(A = B)) & empty(B))) <=> ((~empty(B)) | (A = B) | (~empty(A)))))),
% 0.23/0.43 inference(bind,[status(th)],[])).
% 0.23/0.43 tff(47,plain,
% 0.23/0.43 (![A: $i, B: $i] : (~(empty(A) & (~(A = B)) & empty(B))) <=> ![A: $i, B: $i] : ((~empty(B)) | (A = B) | (~empty(A)))),
% 0.23/0.43 inference(quant_intro,[status(thm)],[46])).
% 0.23/0.43 tff(48,plain,
% 0.23/0.43 (![A: $i, B: $i] : (~(empty(A) & (~(A = B)) & empty(B))) <=> ![A: $i, B: $i] : (~(empty(A) & (~(A = B)) & empty(B)))),
% 0.23/0.43 inference(rewrite,[status(thm)],[])).
% 0.23/0.43 tff(49,plain,
% 0.23/0.43 (^[A: $i, B: $i] : rewrite((~((empty(A) & (~(A = B))) & empty(B))) <=> (~(empty(A) & (~(A = B)) & empty(B))))),
% 0.23/0.43 inference(bind,[status(th)],[])).
% 0.23/0.43 tff(50,plain,
% 0.23/0.43 (![A: $i, B: $i] : (~((empty(A) & (~(A = B))) & empty(B))) <=> ![A: $i, B: $i] : (~(empty(A) & (~(A = B)) & empty(B)))),
% 0.23/0.43 inference(quant_intro,[status(thm)],[49])).
% 0.23/0.43 tff(51,axiom,(![A: $i, B: $i] : (~((empty(A) & (~(A = B))) & empty(B)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t8_boole')).
% 0.23/0.43 tff(52,plain,
% 0.23/0.43 (![A: $i, B: $i] : (~(empty(A) & (~(A = B)) & empty(B)))),
% 0.23/0.43 inference(modus_ponens,[status(thm)],[51, 50])).
% 0.23/0.43 tff(53,plain,
% 0.23/0.43 (![A: $i, B: $i] : (~(empty(A) & (~(A = B)) & empty(B)))),
% 0.23/0.43 inference(modus_ponens,[status(thm)],[52, 48])).
% 0.23/0.43 tff(54,plain,(
% 0.23/0.43 ![A: $i, B: $i] : (~(empty(A) & (~(A = B)) & empty(B)))),
% 0.23/0.43 inference(skolemize,[status(sab)],[53])).
% 0.23/0.43 tff(55,plain,
% 0.23/0.43 (![A: $i, B: $i] : ((~empty(B)) | (A = B) | (~empty(A)))),
% 0.23/0.43 inference(modus_ponens,[status(thm)],[54, 47])).
% 0.23/0.43 tff(56,plain,
% 0.23/0.43 (![A: $i, B: $i] : ((~empty(B)) | (A = B) | (~empty(A)))),
% 0.23/0.43 inference(modus_ponens,[status(thm)],[55, 45])).
% 0.23/0.43 tff(57,plain,
% 0.23/0.43 (((~![A: $i, B: $i] : ((~empty(B)) | (A = B) | (~empty(A)))) | ((~empty(relation_dom(empty_set))) | (~empty(relation_rng(empty_set))) | (relation_dom(empty_set) = relation_rng(empty_set)))) <=> ((~![A: $i, B: $i] : ((~empty(B)) | (A = B) | (~empty(A)))) | (~empty(relation_dom(empty_set))) | (~empty(relation_rng(empty_set))) | (relation_dom(empty_set) = relation_rng(empty_set)))),
% 0.23/0.43 inference(rewrite,[status(thm)],[])).
% 0.23/0.43 tff(58,plain,
% 0.23/0.43 (((~empty(relation_rng(empty_set))) | (relation_dom(empty_set) = relation_rng(empty_set)) | (~empty(relation_dom(empty_set)))) <=> ((~empty(relation_dom(empty_set))) | (~empty(relation_rng(empty_set))) | (relation_dom(empty_set) = relation_rng(empty_set)))),
% 0.23/0.43 inference(rewrite,[status(thm)],[])).
% 0.23/0.43 tff(59,plain,
% 0.23/0.43 (((~![A: $i, B: $i] : ((~empty(B)) | (A = B) | (~empty(A)))) | ((~empty(relation_rng(empty_set))) | (relation_dom(empty_set) = relation_rng(empty_set)) | (~empty(relation_dom(empty_set))))) <=> ((~![A: $i, B: $i] : ((~empty(B)) | (A = B) | (~empty(A)))) | ((~empty(relation_dom(empty_set))) | (~empty(relation_rng(empty_set))) | (relation_dom(empty_set) = relation_rng(empty_set))))),
% 0.23/0.43 inference(monotonicity,[status(thm)],[58])).
% 0.23/0.43 tff(60,plain,
% 0.23/0.43 (((~![A: $i, B: $i] : ((~empty(B)) | (A = B) | (~empty(A)))) | ((~empty(relation_rng(empty_set))) | (relation_dom(empty_set) = relation_rng(empty_set)) | (~empty(relation_dom(empty_set))))) <=> ((~![A: $i, B: $i] : ((~empty(B)) | (A = B) | (~empty(A)))) | (~empty(relation_dom(empty_set))) | (~empty(relation_rng(empty_set))) | (relation_dom(empty_set) = relation_rng(empty_set)))),
% 0.23/0.43 inference(transitivity,[status(thm)],[59, 57])).
% 0.23/0.43 tff(61,plain,
% 0.23/0.43 ((~![A: $i, B: $i] : ((~empty(B)) | (A = B) | (~empty(A)))) | ((~empty(relation_rng(empty_set))) | (relation_dom(empty_set) = relation_rng(empty_set)) | (~empty(relation_dom(empty_set))))),
% 0.23/0.43 inference(quant_inst,[status(thm)],[])).
% 0.23/0.43 tff(62,plain,
% 0.23/0.43 ((~![A: $i, B: $i] : ((~empty(B)) | (A = B) | (~empty(A)))) | (~empty(relation_dom(empty_set))) | (~empty(relation_rng(empty_set))) | (relation_dom(empty_set) = relation_rng(empty_set))),
% 0.23/0.43 inference(modus_ponens,[status(thm)],[61, 60])).
% 0.23/0.43 tff(63,plain,
% 0.23/0.43 (relation_dom(empty_set) = relation_rng(empty_set)),
% 0.23/0.43 inference(unit_resolution,[status(thm)],[62, 56, 43, 24])).
% 0.23/0.43 tff(64,plain,
% 0.23/0.43 (((~![A: $i, B: $i] : ((~empty(B)) | (A = B) | (~empty(A)))) | ((~empty(empty_set)) | (relation_dom(empty_set) = empty_set) | (~empty(relation_dom(empty_set))))) <=> ((~![A: $i, B: $i] : ((~empty(B)) | (A = B) | (~empty(A)))) | (~empty(empty_set)) | (relation_dom(empty_set) = empty_set) | (~empty(relation_dom(empty_set))))),
% 0.23/0.43 inference(rewrite,[status(thm)],[])).
% 0.23/0.43 tff(65,plain,
% 0.23/0.43 ((~![A: $i, B: $i] : ((~empty(B)) | (A = B) | (~empty(A)))) | ((~empty(empty_set)) | (relation_dom(empty_set) = empty_set) | (~empty(relation_dom(empty_set))))),
% 0.23/0.43 inference(quant_inst,[status(thm)],[])).
% 0.23/0.43 tff(66,plain,
% 0.23/0.43 ((~![A: $i, B: $i] : ((~empty(B)) | (A = B) | (~empty(A)))) | (~empty(empty_set)) | (relation_dom(empty_set) = empty_set) | (~empty(relation_dom(empty_set)))),
% 0.23/0.43 inference(modus_ponens,[status(thm)],[65, 64])).
% 0.23/0.43 tff(67,plain,
% 0.23/0.43 (relation_dom(empty_set) = empty_set),
% 0.23/0.43 inference(unit_resolution,[status(thm)],[66, 18, 56, 43])).
% 0.23/0.43 tff(68,plain,
% 0.23/0.43 (empty_set = relation_dom(empty_set)),
% 0.23/0.43 inference(symmetry,[status(thm)],[67])).
% 0.23/0.43 tff(69,plain,
% 0.23/0.43 (empty_set = relation_rng(empty_set)),
% 0.23/0.43 inference(transitivity,[status(thm)],[68, 63])).
% 0.23/0.43 tff(70,plain,
% 0.23/0.43 (subset(empty_set, A!19) <=> subset(relation_rng(empty_set), A!19)),
% 0.23/0.43 inference(monotonicity,[status(thm)],[69])).
% 0.23/0.43 tff(71,plain,
% 0.23/0.43 (subset(relation_rng(empty_set), A!19) <=> subset(empty_set, A!19)),
% 0.23/0.43 inference(symmetry,[status(thm)],[70])).
% 0.23/0.43 tff(72,plain,
% 0.23/0.43 ((~subset(relation_rng(empty_set), A!19)) <=> (~subset(empty_set, A!19))),
% 0.23/0.43 inference(monotonicity,[status(thm)],[71])).
% 0.23/0.43 tff(73,plain,
% 0.23/0.43 (function(empty_set) <=> function(empty_set)),
% 0.23/0.43 inference(rewrite,[status(thm)],[])).
% 0.23/0.43 tff(74,axiom,(((((((relation(empty_set) & relation_empty_yielding(empty_set)) & function(empty_set)) & one_to_one(empty_set)) & empty(empty_set)) & epsilon_transitive(empty_set)) & epsilon_connected(empty_set)) & ordinal(empty_set)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','fc2_ordinal1')).
% 0.23/0.43 tff(75,plain,
% 0.23/0.43 ((((((relation(empty_set) & relation_empty_yielding(empty_set)) & function(empty_set)) & one_to_one(empty_set)) & empty(empty_set)) & epsilon_transitive(empty_set)) & epsilon_connected(empty_set)),
% 0.23/0.43 inference(and_elim,[status(thm)],[74])).
% 0.23/0.43 tff(76,plain,
% 0.23/0.43 (((((relation(empty_set) & relation_empty_yielding(empty_set)) & function(empty_set)) & one_to_one(empty_set)) & empty(empty_set)) & epsilon_transitive(empty_set)),
% 0.23/0.43 inference(and_elim,[status(thm)],[75])).
% 0.23/0.43 tff(77,plain,
% 0.23/0.43 ((((relation(empty_set) & relation_empty_yielding(empty_set)) & function(empty_set)) & one_to_one(empty_set)) & empty(empty_set)),
% 0.23/0.43 inference(and_elim,[status(thm)],[76])).
% 0.23/0.43 tff(78,plain,
% 0.23/0.43 (((relation(empty_set) & relation_empty_yielding(empty_set)) & function(empty_set)) & one_to_one(empty_set)),
% 0.23/0.43 inference(and_elim,[status(thm)],[77])).
% 0.23/0.43 tff(79,plain,
% 0.23/0.43 ((relation(empty_set) & relation_empty_yielding(empty_set)) & function(empty_set)),
% 0.23/0.43 inference(and_elim,[status(thm)],[78])).
% 0.23/0.43 tff(80,plain,
% 0.23/0.43 (function(empty_set)),
% 0.23/0.43 inference(and_elim,[status(thm)],[79])).
% 0.23/0.43 tff(81,plain,
% 0.23/0.43 (function(empty_set)),
% 0.23/0.43 inference(modus_ponens,[status(thm)],[80, 73])).
% 0.23/0.43 tff(82,plain,
% 0.23/0.43 (relation(empty_set) <=> relation(empty_set)),
% 0.23/0.43 inference(rewrite,[status(thm)],[])).
% 0.23/0.43 tff(83,plain,
% 0.23/0.43 (relation(empty_set)),
% 0.23/0.43 inference(and_elim,[status(thm)],[16])).
% 0.23/0.43 tff(84,plain,
% 0.23/0.43 (relation(empty_set)),
% 0.23/0.43 inference(modus_ponens,[status(thm)],[83, 82])).
% 0.23/0.43 tff(85,plain,
% 0.23/0.43 (^[A: $i] : refl(((transfinite_sequence(A) <=> ordinal(relation_dom(A))) | (~relation(A)) | (~function(A))) <=> ((transfinite_sequence(A) <=> ordinal(relation_dom(A))) | (~relation(A)) | (~function(A))))),
% 0.23/0.43 inference(bind,[status(th)],[])).
% 0.23/0.43 tff(86,plain,
% 0.23/0.43 (![A: $i] : ((transfinite_sequence(A) <=> ordinal(relation_dom(A))) | (~relation(A)) | (~function(A))) <=> ![A: $i] : ((transfinite_sequence(A) <=> ordinal(relation_dom(A))) | (~relation(A)) | (~function(A)))),
% 0.23/0.43 inference(quant_intro,[status(thm)],[85])).
% 0.23/0.43 tff(87,plain,
% 0.23/0.43 (^[A: $i] : trans(monotonicity(trans(monotonicity(rewrite((relation(A) & function(A)) <=> (~((~relation(A)) | (~function(A))))), ((~(relation(A) & function(A))) <=> (~(~((~relation(A)) | (~function(A))))))), rewrite((~(~((~relation(A)) | (~function(A))))) <=> ((~relation(A)) | (~function(A)))), ((~(relation(A) & function(A))) <=> ((~relation(A)) | (~function(A))))), (((~(relation(A) & function(A))) | (transfinite_sequence(A) <=> ordinal(relation_dom(A)))) <=> (((~relation(A)) | (~function(A))) | (transfinite_sequence(A) <=> ordinal(relation_dom(A)))))), rewrite((((~relation(A)) | (~function(A))) | (transfinite_sequence(A) <=> ordinal(relation_dom(A)))) <=> ((transfinite_sequence(A) <=> ordinal(relation_dom(A))) | (~relation(A)) | (~function(A)))), (((~(relation(A) & function(A))) | (transfinite_sequence(A) <=> ordinal(relation_dom(A)))) <=> ((transfinite_sequence(A) <=> ordinal(relation_dom(A))) | (~relation(A)) | (~function(A)))))),
% 0.23/0.43 inference(bind,[status(th)],[])).
% 0.23/0.43 tff(88,plain,
% 0.23/0.43 (![A: $i] : ((~(relation(A) & function(A))) | (transfinite_sequence(A) <=> ordinal(relation_dom(A)))) <=> ![A: $i] : ((transfinite_sequence(A) <=> ordinal(relation_dom(A))) | (~relation(A)) | (~function(A)))),
% 0.23/0.43 inference(quant_intro,[status(thm)],[87])).
% 0.23/0.43 tff(89,plain,
% 0.23/0.43 (![A: $i] : ((~(relation(A) & function(A))) | (transfinite_sequence(A) <=> ordinal(relation_dom(A)))) <=> ![A: $i] : ((~(relation(A) & function(A))) | (transfinite_sequence(A) <=> ordinal(relation_dom(A))))),
% 0.23/0.43 inference(rewrite,[status(thm)],[])).
% 0.23/0.43 tff(90,plain,
% 0.23/0.43 (^[A: $i] : rewrite(((relation(A) & function(A)) => (transfinite_sequence(A) <=> ordinal(relation_dom(A)))) <=> ((~(relation(A) & function(A))) | (transfinite_sequence(A) <=> ordinal(relation_dom(A)))))),
% 0.23/0.43 inference(bind,[status(th)],[])).
% 0.23/0.43 tff(91,plain,
% 0.23/0.43 (![A: $i] : ((relation(A) & function(A)) => (transfinite_sequence(A) <=> ordinal(relation_dom(A)))) <=> ![A: $i] : ((~(relation(A) & function(A))) | (transfinite_sequence(A) <=> ordinal(relation_dom(A))))),
% 0.23/0.43 inference(quant_intro,[status(thm)],[90])).
% 0.23/0.43 tff(92,axiom,(![A: $i] : ((relation(A) & function(A)) => (transfinite_sequence(A) <=> ordinal(relation_dom(A))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','d7_ordinal1')).
% 0.23/0.43 tff(93,plain,
% 0.23/0.43 (![A: $i] : ((~(relation(A) & function(A))) | (transfinite_sequence(A) <=> ordinal(relation_dom(A))))),
% 0.23/0.43 inference(modus_ponens,[status(thm)],[92, 91])).
% 0.23/0.43 tff(94,plain,
% 0.23/0.43 (![A: $i] : ((~(relation(A) & function(A))) | (transfinite_sequence(A) <=> ordinal(relation_dom(A))))),
% 0.23/0.43 inference(modus_ponens,[status(thm)],[93, 89])).
% 0.23/0.43 tff(95,plain,(
% 0.23/0.43 ![A: $i] : ((~(relation(A) & function(A))) | (transfinite_sequence(A) <=> ordinal(relation_dom(A))))),
% 0.23/0.43 inference(skolemize,[status(sab)],[94])).
% 0.23/0.43 tff(96,plain,
% 0.23/0.43 (![A: $i] : ((transfinite_sequence(A) <=> ordinal(relation_dom(A))) | (~relation(A)) | (~function(A)))),
% 0.23/0.43 inference(modus_ponens,[status(thm)],[95, 88])).
% 0.23/0.43 tff(97,plain,
% 0.23/0.43 (![A: $i] : ((transfinite_sequence(A) <=> ordinal(relation_dom(A))) | (~relation(A)) | (~function(A)))),
% 0.23/0.43 inference(modus_ponens,[status(thm)],[96, 86])).
% 0.23/0.43 tff(98,plain,
% 0.23/0.43 (((~![A: $i] : ((transfinite_sequence(A) <=> ordinal(relation_dom(A))) | (~relation(A)) | (~function(A)))) | ((~relation(empty_set)) | (~function(empty_set)) | (transfinite_sequence(empty_set) <=> ordinal(relation_dom(empty_set))))) <=> ((~![A: $i] : ((transfinite_sequence(A) <=> ordinal(relation_dom(A))) | (~relation(A)) | (~function(A)))) | (~relation(empty_set)) | (~function(empty_set)) | (transfinite_sequence(empty_set) <=> ordinal(relation_dom(empty_set))))),
% 0.23/0.43 inference(rewrite,[status(thm)],[])).
% 0.23/0.43 tff(99,plain,
% 0.23/0.43 (((transfinite_sequence(empty_set) <=> ordinal(relation_dom(empty_set))) | (~relation(empty_set)) | (~function(empty_set))) <=> ((~relation(empty_set)) | (~function(empty_set)) | (transfinite_sequence(empty_set) <=> ordinal(relation_dom(empty_set))))),
% 0.23/0.43 inference(rewrite,[status(thm)],[])).
% 0.23/0.43 tff(100,plain,
% 0.23/0.43 (((~![A: $i] : ((transfinite_sequence(A) <=> ordinal(relation_dom(A))) | (~relation(A)) | (~function(A)))) | ((transfinite_sequence(empty_set) <=> ordinal(relation_dom(empty_set))) | (~relation(empty_set)) | (~function(empty_set)))) <=> ((~![A: $i] : ((transfinite_sequence(A) <=> ordinal(relation_dom(A))) | (~relation(A)) | (~function(A)))) | ((~relation(empty_set)) | (~function(empty_set)) | (transfinite_sequence(empty_set) <=> ordinal(relation_dom(empty_set)))))),
% 0.23/0.43 inference(monotonicity,[status(thm)],[99])).
% 0.23/0.43 tff(101,plain,
% 0.23/0.43 (((~![A: $i] : ((transfinite_sequence(A) <=> ordinal(relation_dom(A))) | (~relation(A)) | (~function(A)))) | ((transfinite_sequence(empty_set) <=> ordinal(relation_dom(empty_set))) | (~relation(empty_set)) | (~function(empty_set)))) <=> ((~![A: $i] : ((transfinite_sequence(A) <=> ordinal(relation_dom(A))) | (~relation(A)) | (~function(A)))) | (~relation(empty_set)) | (~function(empty_set)) | (transfinite_sequence(empty_set) <=> ordinal(relation_dom(empty_set))))),
% 0.23/0.43 inference(transitivity,[status(thm)],[100, 98])).
% 0.23/0.43 tff(102,plain,
% 0.23/0.43 ((~![A: $i] : ((transfinite_sequence(A) <=> ordinal(relation_dom(A))) | (~relation(A)) | (~function(A)))) | ((transfinite_sequence(empty_set) <=> ordinal(relation_dom(empty_set))) | (~relation(empty_set)) | (~function(empty_set)))),
% 0.23/0.43 inference(quant_inst,[status(thm)],[])).
% 0.23/0.43 tff(103,plain,
% 0.23/0.43 ((~![A: $i] : ((transfinite_sequence(A) <=> ordinal(relation_dom(A))) | (~relation(A)) | (~function(A)))) | (~relation(empty_set)) | (~function(empty_set)) | (transfinite_sequence(empty_set) <=> ordinal(relation_dom(empty_set)))),
% 0.23/0.43 inference(modus_ponens,[status(thm)],[102, 101])).
% 0.23/0.43 tff(104,plain,
% 0.23/0.43 (transfinite_sequence(empty_set) <=> ordinal(relation_dom(empty_set))),
% 0.23/0.43 inference(unit_resolution,[status(thm)],[103, 97, 84, 81])).
% 0.23/0.43 tff(105,plain,
% 0.23/0.43 (^[A: $i] : refl(((~empty(A)) | (~((~ordinal(A)) | (~epsilon_transitive(A)) | (~epsilon_connected(A))))) <=> ((~empty(A)) | (~((~ordinal(A)) | (~epsilon_transitive(A)) | (~epsilon_connected(A))))))),
% 0.23/0.43 inference(bind,[status(th)],[])).
% 0.23/0.43 tff(106,plain,
% 0.23/0.43 (![A: $i] : ((~empty(A)) | (~((~ordinal(A)) | (~epsilon_transitive(A)) | (~epsilon_connected(A))))) <=> ![A: $i] : ((~empty(A)) | (~((~ordinal(A)) | (~epsilon_transitive(A)) | (~epsilon_connected(A)))))),
% 0.23/0.43 inference(quant_intro,[status(thm)],[105])).
% 0.23/0.43 tff(107,plain,
% 0.23/0.43 (^[A: $i] : rewrite(((~empty(A)) | (epsilon_transitive(A) & epsilon_connected(A) & ordinal(A))) <=> ((~empty(A)) | (~((~ordinal(A)) | (~epsilon_transitive(A)) | (~epsilon_connected(A))))))),
% 0.23/0.43 inference(bind,[status(th)],[])).
% 0.23/0.43 tff(108,plain,
% 0.23/0.43 (![A: $i] : ((~empty(A)) | (epsilon_transitive(A) & epsilon_connected(A) & ordinal(A))) <=> ![A: $i] : ((~empty(A)) | (~((~ordinal(A)) | (~epsilon_transitive(A)) | (~epsilon_connected(A)))))),
% 0.23/0.43 inference(quant_intro,[status(thm)],[107])).
% 0.23/0.43 tff(109,plain,
% 0.23/0.43 (![A: $i] : ((~empty(A)) | (epsilon_transitive(A) & epsilon_connected(A) & ordinal(A))) <=> ![A: $i] : ((~empty(A)) | (epsilon_transitive(A) & epsilon_connected(A) & ordinal(A)))),
% 0.23/0.43 inference(rewrite,[status(thm)],[])).
% 0.23/0.43 tff(110,plain,
% 0.23/0.43 (^[A: $i] : trans(monotonicity(rewrite(((epsilon_transitive(A) & epsilon_connected(A)) & ordinal(A)) <=> (epsilon_transitive(A) & epsilon_connected(A) & ordinal(A))), ((empty(A) => ((epsilon_transitive(A) & epsilon_connected(A)) & ordinal(A))) <=> (empty(A) => (epsilon_transitive(A) & epsilon_connected(A) & ordinal(A))))), rewrite((empty(A) => (epsilon_transitive(A) & epsilon_connected(A) & ordinal(A))) <=> ((~empty(A)) | (epsilon_transitive(A) & epsilon_connected(A) & ordinal(A)))), ((empty(A) => ((epsilon_transitive(A) & epsilon_connected(A)) & ordinal(A))) <=> ((~empty(A)) | (epsilon_transitive(A) & epsilon_connected(A) & ordinal(A)))))),
% 0.23/0.43 inference(bind,[status(th)],[])).
% 0.23/0.43 tff(111,plain,
% 0.23/0.43 (![A: $i] : (empty(A) => ((epsilon_transitive(A) & epsilon_connected(A)) & ordinal(A))) <=> ![A: $i] : ((~empty(A)) | (epsilon_transitive(A) & epsilon_connected(A) & ordinal(A)))),
% 0.23/0.43 inference(quant_intro,[status(thm)],[110])).
% 0.23/0.43 tff(112,axiom,(![A: $i] : (empty(A) => ((epsilon_transitive(A) & epsilon_connected(A)) & ordinal(A)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','cc3_ordinal1')).
% 0.23/0.43 tff(113,plain,
% 0.23/0.43 (![A: $i] : ((~empty(A)) | (epsilon_transitive(A) & epsilon_connected(A) & ordinal(A)))),
% 0.23/0.43 inference(modus_ponens,[status(thm)],[112, 111])).
% 0.23/0.43 tff(114,plain,
% 0.23/0.43 (![A: $i] : ((~empty(A)) | (epsilon_transitive(A) & epsilon_connected(A) & ordinal(A)))),
% 0.23/0.43 inference(modus_ponens,[status(thm)],[113, 109])).
% 0.23/0.43 tff(115,plain,(
% 0.23/0.43 ![A: $i] : ((~empty(A)) | (epsilon_transitive(A) & epsilon_connected(A) & ordinal(A)))),
% 0.23/0.43 inference(skolemize,[status(sab)],[114])).
% 0.23/0.43 tff(116,plain,
% 0.23/0.43 (![A: $i] : ((~empty(A)) | (~((~ordinal(A)) | (~epsilon_transitive(A)) | (~epsilon_connected(A)))))),
% 0.23/0.43 inference(modus_ponens,[status(thm)],[115, 108])).
% 0.23/0.43 tff(117,plain,
% 0.23/0.43 (![A: $i] : ((~empty(A)) | (~((~ordinal(A)) | (~epsilon_transitive(A)) | (~epsilon_connected(A)))))),
% 0.23/0.43 inference(modus_ponens,[status(thm)],[116, 106])).
% 0.23/0.43 tff(118,plain,
% 0.23/0.43 (((~![A: $i] : ((~empty(A)) | (~((~ordinal(A)) | (~epsilon_transitive(A)) | (~epsilon_connected(A)))))) | ((~empty(relation_dom(empty_set))) | (~((~ordinal(relation_dom(empty_set))) | (~epsilon_transitive(relation_dom(empty_set))) | (~epsilon_connected(relation_dom(empty_set))))))) <=> ((~![A: $i] : ((~empty(A)) | (~((~ordinal(A)) | (~epsilon_transitive(A)) | (~epsilon_connected(A)))))) | (~empty(relation_dom(empty_set))) | (~((~ordinal(relation_dom(empty_set))) | (~epsilon_transitive(relation_dom(empty_set))) | (~epsilon_connected(relation_dom(empty_set))))))),
% 0.23/0.43 inference(rewrite,[status(thm)],[])).
% 0.23/0.43 tff(119,plain,
% 0.23/0.43 ((~![A: $i] : ((~empty(A)) | (~((~ordinal(A)) | (~epsilon_transitive(A)) | (~epsilon_connected(A)))))) | ((~empty(relation_dom(empty_set))) | (~((~ordinal(relation_dom(empty_set))) | (~epsilon_transitive(relation_dom(empty_set))) | (~epsilon_connected(relation_dom(empty_set))))))),
% 0.23/0.43 inference(quant_inst,[status(thm)],[])).
% 0.23/0.43 tff(120,plain,
% 0.23/0.43 ((~![A: $i] : ((~empty(A)) | (~((~ordinal(A)) | (~epsilon_transitive(A)) | (~epsilon_connected(A)))))) | (~empty(relation_dom(empty_set))) | (~((~ordinal(relation_dom(empty_set))) | (~epsilon_transitive(relation_dom(empty_set))) | (~epsilon_connected(relation_dom(empty_set)))))),
% 0.23/0.43 inference(modus_ponens,[status(thm)],[119, 118])).
% 0.23/0.43 tff(121,plain,
% 0.23/0.43 ((~empty(relation_dom(empty_set))) | (~((~ordinal(relation_dom(empty_set))) | (~epsilon_transitive(relation_dom(empty_set))) | (~epsilon_connected(relation_dom(empty_set)))))),
% 0.23/0.43 inference(unit_resolution,[status(thm)],[120, 117])).
% 0.23/0.43 tff(122,plain,
% 0.23/0.43 (~((~ordinal(relation_dom(empty_set))) | (~epsilon_transitive(relation_dom(empty_set))) | (~epsilon_connected(relation_dom(empty_set))))),
% 0.23/0.43 inference(unit_resolution,[status(thm)],[121, 43])).
% 0.23/0.43 tff(123,plain,
% 0.23/0.43 (((~ordinal(relation_dom(empty_set))) | (~epsilon_transitive(relation_dom(empty_set))) | (~epsilon_connected(relation_dom(empty_set)))) | ordinal(relation_dom(empty_set))),
% 0.23/0.43 inference(tautology,[status(thm)],[])).
% 0.23/0.43 tff(124,plain,
% 0.23/0.43 (ordinal(relation_dom(empty_set))),
% 0.23/0.43 inference(unit_resolution,[status(thm)],[123, 122])).
% 0.23/0.43 tff(125,plain,
% 0.23/0.43 ((~(transfinite_sequence(empty_set) <=> ordinal(relation_dom(empty_set)))) | transfinite_sequence(empty_set) | (~ordinal(relation_dom(empty_set)))),
% 0.23/0.43 inference(tautology,[status(thm)],[])).
% 0.23/0.43 tff(126,plain,
% 0.23/0.43 (transfinite_sequence(empty_set)),
% 0.23/0.43 inference(unit_resolution,[status(thm)],[125, 124, 104])).
% 0.23/0.43 tff(127,plain,
% 0.23/0.43 (^[A: $i, B: $i] : refl(((transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)) | (~relation(B)) | (~function(B)) | (~transfinite_sequence(B))) <=> ((transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)) | (~relation(B)) | (~function(B)) | (~transfinite_sequence(B))))),
% 0.23/0.43 inference(bind,[status(th)],[])).
% 0.23/0.43 tff(128,plain,
% 0.23/0.43 (![A: $i, B: $i] : ((transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)) | (~relation(B)) | (~function(B)) | (~transfinite_sequence(B))) <=> ![A: $i, B: $i] : ((transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)) | (~relation(B)) | (~function(B)) | (~transfinite_sequence(B)))),
% 0.23/0.43 inference(quant_intro,[status(thm)],[127])).
% 0.23/0.43 tff(129,plain,
% 0.23/0.43 (^[A: $i, B: $i] : trans(monotonicity(trans(monotonicity(rewrite((relation(B) & function(B) & transfinite_sequence(B)) <=> (~((~relation(B)) | (~function(B)) | (~transfinite_sequence(B))))), ((~(relation(B) & function(B) & transfinite_sequence(B))) <=> (~(~((~relation(B)) | (~function(B)) | (~transfinite_sequence(B))))))), rewrite((~(~((~relation(B)) | (~function(B)) | (~transfinite_sequence(B))))) <=> ((~relation(B)) | (~function(B)) | (~transfinite_sequence(B)))), ((~(relation(B) & function(B) & transfinite_sequence(B))) <=> ((~relation(B)) | (~function(B)) | (~transfinite_sequence(B))))), (((~(relation(B) & function(B) & transfinite_sequence(B))) | (transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A))) <=> (((~relation(B)) | (~function(B)) | (~transfinite_sequence(B))) | (transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A))))), rewrite((((~relation(B)) | (~function(B)) | (~transfinite_sequence(B))) | (transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A))) <=> ((transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)) | (~relation(B)) | (~function(B)) | (~transfinite_sequence(B)))), (((~(relation(B) & function(B) & transfinite_sequence(B))) | (transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A))) <=> ((transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)) | (~relation(B)) | (~function(B)) | (~transfinite_sequence(B)))))),
% 0.23/0.43 inference(bind,[status(th)],[])).
% 0.23/0.43 tff(130,plain,
% 0.23/0.43 (![A: $i, B: $i] : ((~(relation(B) & function(B) & transfinite_sequence(B))) | (transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A))) <=> ![A: $i, B: $i] : ((transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)) | (~relation(B)) | (~function(B)) | (~transfinite_sequence(B)))),
% 0.23/0.43 inference(quant_intro,[status(thm)],[129])).
% 0.23/0.43 tff(131,plain,
% 0.23/0.43 (![A: $i, B: $i] : ((~(relation(B) & function(B) & transfinite_sequence(B))) | (transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A))) <=> ![A: $i, B: $i] : ((~(relation(B) & function(B) & transfinite_sequence(B))) | (transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)))),
% 0.23/0.43 inference(rewrite,[status(thm)],[])).
% 0.23/0.43 tff(132,plain,
% 0.23/0.43 (^[A: $i, B: $i] : trans(monotonicity(rewrite(((relation(B) & function(B)) & transfinite_sequence(B)) <=> (relation(B) & function(B) & transfinite_sequence(B))), ((((relation(B) & function(B)) & transfinite_sequence(B)) => (transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A))) <=> ((relation(B) & function(B) & transfinite_sequence(B)) => (transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A))))), rewrite(((relation(B) & function(B) & transfinite_sequence(B)) => (transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A))) <=> ((~(relation(B) & function(B) & transfinite_sequence(B))) | (transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)))), ((((relation(B) & function(B)) & transfinite_sequence(B)) => (transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A))) <=> ((~(relation(B) & function(B) & transfinite_sequence(B))) | (transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)))))),
% 0.23/0.44 inference(bind,[status(th)],[])).
% 0.23/0.44 tff(133,plain,
% 0.23/0.44 (![A: $i, B: $i] : (((relation(B) & function(B)) & transfinite_sequence(B)) => (transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A))) <=> ![A: $i, B: $i] : ((~(relation(B) & function(B) & transfinite_sequence(B))) | (transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)))),
% 0.23/0.44 inference(quant_intro,[status(thm)],[132])).
% 0.23/0.44 tff(134,axiom,(![A: $i, B: $i] : (((relation(B) & function(B)) & transfinite_sequence(B)) => (transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','d8_ordinal1')).
% 0.23/0.44 tff(135,plain,
% 0.23/0.44 (![A: $i, B: $i] : ((~(relation(B) & function(B) & transfinite_sequence(B))) | (transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)))),
% 0.23/0.44 inference(modus_ponens,[status(thm)],[134, 133])).
% 0.23/0.44 tff(136,plain,
% 0.23/0.44 (![A: $i, B: $i] : ((~(relation(B) & function(B) & transfinite_sequence(B))) | (transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)))),
% 0.23/0.44 inference(modus_ponens,[status(thm)],[135, 131])).
% 0.23/0.44 tff(137,plain,(
% 0.23/0.44 ![A: $i, B: $i] : ((~(relation(B) & function(B) & transfinite_sequence(B))) | (transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)))),
% 0.23/0.44 inference(skolemize,[status(sab)],[136])).
% 0.23/0.44 tff(138,plain,
% 0.23/0.44 (![A: $i, B: $i] : ((transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)) | (~relation(B)) | (~function(B)) | (~transfinite_sequence(B)))),
% 0.23/0.44 inference(modus_ponens,[status(thm)],[137, 130])).
% 0.23/0.44 tff(139,plain,
% 0.23/0.44 (![A: $i, B: $i] : ((transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)) | (~relation(B)) | (~function(B)) | (~transfinite_sequence(B)))),
% 0.23/0.44 inference(modus_ponens,[status(thm)],[138, 128])).
% 0.23/0.44 tff(140,plain,
% 0.23/0.44 (((~![A: $i, B: $i] : ((transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)) | (~relation(B)) | (~function(B)) | (~transfinite_sequence(B)))) | ((~relation(empty_set)) | (~function(empty_set)) | (~transfinite_sequence(empty_set)) | (transfinite_sequence_of(empty_set, A!19) <=> subset(relation_rng(empty_set), A!19)))) <=> ((~![A: $i, B: $i] : ((transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)) | (~relation(B)) | (~function(B)) | (~transfinite_sequence(B)))) | (~relation(empty_set)) | (~function(empty_set)) | (~transfinite_sequence(empty_set)) | (transfinite_sequence_of(empty_set, A!19) <=> subset(relation_rng(empty_set), A!19)))),
% 0.23/0.44 inference(rewrite,[status(thm)],[])).
% 0.23/0.44 tff(141,plain,
% 0.23/0.44 (((transfinite_sequence_of(empty_set, A!19) <=> subset(relation_rng(empty_set), A!19)) | (~relation(empty_set)) | (~function(empty_set)) | (~transfinite_sequence(empty_set))) <=> ((~relation(empty_set)) | (~function(empty_set)) | (~transfinite_sequence(empty_set)) | (transfinite_sequence_of(empty_set, A!19) <=> subset(relation_rng(empty_set), A!19)))),
% 0.23/0.44 inference(rewrite,[status(thm)],[])).
% 0.23/0.44 tff(142,plain,
% 0.23/0.44 (((~![A: $i, B: $i] : ((transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)) | (~relation(B)) | (~function(B)) | (~transfinite_sequence(B)))) | ((transfinite_sequence_of(empty_set, A!19) <=> subset(relation_rng(empty_set), A!19)) | (~relation(empty_set)) | (~function(empty_set)) | (~transfinite_sequence(empty_set)))) <=> ((~![A: $i, B: $i] : ((transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)) | (~relation(B)) | (~function(B)) | (~transfinite_sequence(B)))) | ((~relation(empty_set)) | (~function(empty_set)) | (~transfinite_sequence(empty_set)) | (transfinite_sequence_of(empty_set, A!19) <=> subset(relation_rng(empty_set), A!19))))),
% 0.23/0.44 inference(monotonicity,[status(thm)],[141])).
% 0.23/0.44 tff(143,plain,
% 0.23/0.44 (((~![A: $i, B: $i] : ((transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)) | (~relation(B)) | (~function(B)) | (~transfinite_sequence(B)))) | ((transfinite_sequence_of(empty_set, A!19) <=> subset(relation_rng(empty_set), A!19)) | (~relation(empty_set)) | (~function(empty_set)) | (~transfinite_sequence(empty_set)))) <=> ((~![A: $i, B: $i] : ((transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)) | (~relation(B)) | (~function(B)) | (~transfinite_sequence(B)))) | (~relation(empty_set)) | (~function(empty_set)) | (~transfinite_sequence(empty_set)) | (transfinite_sequence_of(empty_set, A!19) <=> subset(relation_rng(empty_set), A!19)))),
% 0.23/0.44 inference(transitivity,[status(thm)],[142, 140])).
% 0.23/0.44 tff(144,plain,
% 0.23/0.44 ((~![A: $i, B: $i] : ((transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)) | (~relation(B)) | (~function(B)) | (~transfinite_sequence(B)))) | ((transfinite_sequence_of(empty_set, A!19) <=> subset(relation_rng(empty_set), A!19)) | (~relation(empty_set)) | (~function(empty_set)) | (~transfinite_sequence(empty_set)))),
% 0.23/0.44 inference(quant_inst,[status(thm)],[])).
% 0.23/0.44 tff(145,plain,
% 0.23/0.44 ((~![A: $i, B: $i] : ((transfinite_sequence_of(B, A) <=> subset(relation_rng(B), A)) | (~relation(B)) | (~function(B)) | (~transfinite_sequence(B)))) | (~relation(empty_set)) | (~function(empty_set)) | (~transfinite_sequence(empty_set)) | (transfinite_sequence_of(empty_set, A!19) <=> subset(relation_rng(empty_set), A!19))),
% 0.23/0.44 inference(modus_ponens,[status(thm)],[144, 143])).
% 0.23/0.44 tff(146,plain,
% 0.23/0.44 ((~transfinite_sequence(empty_set)) | (transfinite_sequence_of(empty_set, A!19) <=> subset(relation_rng(empty_set), A!19))),
% 0.23/0.44 inference(unit_resolution,[status(thm)],[145, 139, 84, 81])).
% 0.23/0.44 tff(147,plain,
% 0.23/0.44 (transfinite_sequence_of(empty_set, A!19) <=> subset(relation_rng(empty_set), A!19)),
% 0.23/0.44 inference(unit_resolution,[status(thm)],[146, 126])).
% 0.23/0.44 tff(148,plain,
% 0.23/0.44 ((~![A: $i] : transfinite_sequence_of(empty_set, A)) <=> (~![A: $i] : transfinite_sequence_of(empty_set, A))),
% 0.23/0.44 inference(rewrite,[status(thm)],[])).
% 0.23/0.44 tff(149,axiom,(~![A: $i] : transfinite_sequence_of(empty_set, A)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t45_ordinal1')).
% 0.23/0.44 tff(150,plain,
% 0.23/0.44 (~![A: $i] : transfinite_sequence_of(empty_set, A)),
% 0.23/0.44 inference(modus_ponens,[status(thm)],[149, 148])).
% 0.23/0.44 tff(151,plain,
% 0.23/0.44 (~![A: $i] : transfinite_sequence_of(empty_set, A)),
% 0.23/0.44 inference(modus_ponens,[status(thm)],[150, 148])).
% 0.23/0.44 tff(152,plain,
% 0.23/0.44 (~![A: $i] : transfinite_sequence_of(empty_set, A)),
% 0.23/0.44 inference(modus_ponens,[status(thm)],[151, 148])).
% 0.23/0.44 tff(153,plain,
% 0.23/0.44 (~![A: $i] : transfinite_sequence_of(empty_set, A)),
% 0.23/0.44 inference(modus_ponens,[status(thm)],[152, 148])).
% 0.23/0.44 tff(154,plain,(
% 0.23/0.44 ~transfinite_sequence_of(empty_set, A!19)),
% 0.23/0.44 inference(skolemize,[status(sab)],[153])).
% 0.23/0.44 tff(155,plain,
% 0.23/0.44 ((~(transfinite_sequence_of(empty_set, A!19) <=> subset(relation_rng(empty_set), A!19))) | transfinite_sequence_of(empty_set, A!19) | (~subset(relation_rng(empty_set), A!19))),
% 0.23/0.44 inference(tautology,[status(thm)],[])).
% 0.23/0.44 tff(156,plain,
% 0.23/0.44 ((~(transfinite_sequence_of(empty_set, A!19) <=> subset(relation_rng(empty_set), A!19))) | (~subset(relation_rng(empty_set), A!19))),
% 0.23/0.44 inference(unit_resolution,[status(thm)],[155, 154])).
% 0.23/0.44 tff(157,plain,
% 0.23/0.44 (~subset(relation_rng(empty_set), A!19)),
% 0.23/0.44 inference(unit_resolution,[status(thm)],[156, 147])).
% 0.23/0.44 tff(158,plain,
% 0.23/0.44 (~subset(empty_set, A!19)),
% 0.23/0.44 inference(modus_ponens,[status(thm)],[157, 72])).
% 0.23/0.44 tff(159,plain,
% 0.23/0.44 (^[A: $i] : refl(subset(empty_set, A) <=> subset(empty_set, A))),
% 0.23/0.44 inference(bind,[status(th)],[])).
% 0.23/0.44 tff(160,plain,
% 0.23/0.44 (![A: $i] : subset(empty_set, A) <=> ![A: $i] : subset(empty_set, A)),
% 0.23/0.44 inference(quant_intro,[status(thm)],[159])).
% 0.23/0.44 tff(161,plain,
% 0.23/0.44 (![A: $i] : subset(empty_set, A) <=> ![A: $i] : subset(empty_set, A)),
% 0.23/0.44 inference(rewrite,[status(thm)],[])).
% 0.23/0.44 tff(162,axiom,(![A: $i] : subset(empty_set, A)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t2_xboole_1')).
% 0.23/0.44 tff(163,plain,
% 0.23/0.44 (![A: $i] : subset(empty_set, A)),
% 0.23/0.44 inference(modus_ponens,[status(thm)],[162, 161])).
% 0.23/0.44 tff(164,plain,(
% 0.23/0.44 ![A: $i] : subset(empty_set, A)),
% 0.23/0.44 inference(skolemize,[status(sab)],[163])).
% 0.23/0.44 tff(165,plain,
% 0.23/0.44 (![A: $i] : subset(empty_set, A)),
% 0.23/0.44 inference(modus_ponens,[status(thm)],[164, 160])).
% 0.23/0.44 tff(166,plain,
% 0.23/0.44 ((~![A: $i] : subset(empty_set, A)) | subset(empty_set, A!19)),
% 0.23/0.44 inference(quant_inst,[status(thm)],[])).
% 0.23/0.44 tff(167,plain,
% 0.23/0.44 (subset(empty_set, A!19)),
% 0.23/0.44 inference(unit_resolution,[status(thm)],[166, 165])).
% 0.23/0.44 tff(168,plain,
% 0.23/0.44 ($false),
% 0.23/0.44 inference(unit_resolution,[status(thm)],[167, 158])).
% 0.23/0.44 % SZS output end Proof
% 0.23/0.44 % E exiting
%------------------------------------------------------------------------------