0.03/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.03/0.13	% Command    : tptp2X_and_run_prover9 %d %s
0.12/0.34	% Computer   : n004.cluster.edu
0.12/0.34	% Model      : x86_64 x86_64
0.12/0.34	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.12/0.34	% Memory     : 8042.1875MB
0.12/0.34	% OS         : Linux 3.10.0-693.el7.x86_64
0.12/0.34	% CPULimit   : 960
0.12/0.34	% DateTime   : Thu Jul  2 07:43:58 EDT 2020
0.12/0.34	% CPUTime    : 
1.00/1.26	============================== Prover9 ===============================
1.00/1.26	Prover9 (32) version 2009-11A, November 2009.
1.00/1.26	Process 7407 was started by sandbox2 on n004.cluster.edu,
1.00/1.26	Thu Jul  2 07:43:59 2020
1.00/1.26	The command was "/export/starexec/sandbox2/solver/bin/prover9 -t 960 -f /tmp/Prover9_7254_n004.cluster.edu".
1.00/1.26	============================== end of head ===========================
1.00/1.26	
1.00/1.26	============================== INPUT =================================
1.00/1.26	
1.00/1.26	% Reading from file /tmp/Prover9_7254_n004.cluster.edu
1.00/1.26	
1.00/1.26	set(prolog_style_variables).
1.00/1.26	set(auto2).
1.00/1.26	    % set(auto2) -> set(auto).
1.00/1.26	    % set(auto) -> set(auto_inference).
1.00/1.26	    % set(auto) -> set(auto_setup).
1.00/1.26	    % set(auto_setup) -> set(predicate_elim).
1.00/1.26	    % set(auto_setup) -> assign(eq_defs, unfold).
1.00/1.26	    % set(auto) -> set(auto_limits).
1.00/1.26	    % set(auto_limits) -> assign(max_weight, "100.000").
1.00/1.26	    % set(auto_limits) -> assign(sos_limit, 20000).
1.00/1.26	    % set(auto) -> set(auto_denials).
1.00/1.26	    % set(auto) -> set(auto_process).
1.00/1.26	    % set(auto2) -> assign(new_constants, 1).
1.00/1.26	    % set(auto2) -> assign(fold_denial_max, 3).
1.00/1.27	    % set(auto2) -> assign(max_weight, "200.000").
1.00/1.27	    % set(auto2) -> assign(max_hours, 1).
1.00/1.27	    % assign(max_hours, 1) -> assign(max_seconds, 3600).
1.00/1.27	    % set(auto2) -> assign(max_seconds, 0).
1.00/1.27	    % set(auto2) -> assign(max_minutes, 5).
1.00/1.27	    % assign(max_minutes, 5) -> assign(max_seconds, 300).
1.00/1.27	    % set(auto2) -> set(sort_initial_sos).
1.00/1.27	    % set(auto2) -> assign(sos_limit, -1).
1.00/1.27	    % set(auto2) -> assign(lrs_ticks, 3000).
1.00/1.27	    % set(auto2) -> assign(max_megs, 400).
1.00/1.27	    % set(auto2) -> assign(stats, some).
1.00/1.27	    % set(auto2) -> clear(echo_input).
1.00/1.27	    % set(auto2) -> set(quiet).
1.00/1.27	    % set(auto2) -> clear(print_initial_clauses).
1.00/1.27	    % set(auto2) -> clear(print_given).
1.00/1.27	assign(lrs_ticks,-1).
1.00/1.27	assign(sos_limit,10000).
1.00/1.27	assign(order,kbo).
1.00/1.27	set(lex_order_vars).
1.00/1.27	clear(print_given).
1.00/1.27	
1.00/1.27	% formulas(sos).  % not echoed (342 formulas)
1.00/1.27	
1.00/1.27	============================== end of input ==========================
1.00/1.27	
1.00/1.27	% From the command line: assign(max_seconds, 960).
1.00/1.27	
1.00/1.27	============================== PROCESS NON-CLAUSAL FORMULAS ==========
1.00/1.27	
1.00/1.27	% Formulas that are not ordinary clauses:
1.00/1.27	1 (all A (relation(A) -> (all B (relation(B) -> (B = relation_inverse(A) <-> (all C all D (in(ordered_pair(C,D),B) <-> in(ordered_pair(D,C),A)))))))) # label(d7_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	2 (all A exists B ((all C -(in(C,B) & (all D -(in(D,B) & (all E (subset(E,C) -> in(E,D))))))) & (all C -(-are_equipotent(C,B) & -in(C,B) & subset(C,B))) & (all C all D (in(C,B) & subset(D,C) -> in(D,B))) & in(A,B))) # label(t9_tarski) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	3 (all A all B all C (relation(C) -> relation_dom_restriction(relation_rng_restriction(A,C),B) = relation_rng_restriction(A,relation_dom_restriction(C,B)))) # label(t140_relat_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	4 (all A all B (element(A,B) -> in(A,B) | empty(B))) # label(t2_subset) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	5 (all A (relation(A) -> (all B (relation(B) -> subset(relation_dom(relation_composition(A,B)),relation_dom(A)))))) # label(t44_relat_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	6 (all A all B all C -(in(C,A) & in(B,C) & in(A,B))) # label(t3_ordinal1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	7 (all A all B (element(B,powerset(powerset(A))) -> element(meet_of_subsets(A,B),powerset(A)))) # label(dt_k6_setfam_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	8 (all A all B all C (subset(A,B) & disjoint(B,C) -> disjoint(A,C))) # label(t63_xboole_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	9 (all A all B unordered_pair(B,A) = unordered_pair(A,B)) # label(commutativity_k2_tarski) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	10 (all A -empty(succ(A))) # label(fc1_ordinal1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	11 (exists A (epsilon_transitive(A) & epsilon_connected(A) & ordinal(A))) # label(rc1_ordinal1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	12 (all A element(cast_to_subset(A),powerset(A))) # label(dt_k2_subset_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	13 (all A all B (relation(B) -> relation_restriction(B,A) = relation_dom_restriction(relation_rng_restriction(A,B),A))) # label(t17_wellord1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	14 (all A (ordinal(A) -> (all B (ordinal(B) -> -(-in(A,B) & -in(B,A) & B != A))))) # label(t24_ordinal1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	15 (all A ((all B -in(B,A)) <-> A = empty_set)) # label(d1_xboole_0) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	16 (all A all B (function(A) & relation(A) -> relation(relation_dom_restriction(A,B)) & function(relation_dom_restriction(A,B)))) # label(fc4_funct_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	17 (all A all B (function(B) & relation(B) -> (all C (function(C) & relation(C) -> (in(A,relation_dom(B)) -> apply(C,apply(B,A)) = apply(relation_composition(B,C),A)))))) # label(t23_funct_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	18 (all A (function(A) & relation(A) -> (all B ((all C (in(C,B) <-> (exists D (in(D,relation_dom(A)) & apply(A,D) = C)))) <-> B = relation_rng(A))))) # label(d5_funct_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	19 (all A all B (-((all C -in(C,set_intersection2(A,B))) & -disjoint(A,B)) & -(disjoint(A,B) & (exists C in(C,set_intersection2(A,B)))))) # label(t4_xboole_0) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	20 (all A all B (subset(B,A) & subset(A,B) <-> A = B)) # label(d10_xboole_0) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	21 (all A all B all C (subset(A,B) & subset(A,C) -> subset(A,set_intersection2(B,C)))) # label(t19_xboole_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	22 (all A exists B element(B,A)) # label(existence_m1_subset_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	23 (all A (relation(A) & function(A) -> (all B all C ((all D ((exists E (in(E,relation_dom(A)) & apply(A,E) = D & in(E,B))) <-> in(D,C))) <-> relation_image(A,B) = C)))) # label(d12_funct_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	24 (all A all B (relation(A) -> relation(relation_restriction(A,B)))) # label(dt_k2_wellord1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	25 (all A (relation(A) -> ((all B all C all D (in(ordered_pair(B,C),A) & in(ordered_pair(C,D),A) -> in(ordered_pair(B,D),A))) <-> transitive(A)))) # label(l2_wellord1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	26 (all A all B (relation(A) -> relation(relation_dom_restriction(A,B)))) # label(dt_k7_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	27 (all A all B ((all C (in(C,A) -> in(C,B))) -> element(A,powerset(B)))) # label(l71_subset_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	28 (all A all B (subset(A,B) <-> empty_set = set_difference(A,B))) # label(t37_xboole_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	29 (all A A = union(powerset(A))) # label(t99_zfmisc_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	30 (all A (empty(A) -> function(A))) # label(cc1_funct_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	31 (all A (epsilon_transitive(A) <-> (all B (in(B,A) -> subset(B,A))))) # label(d2_ordinal1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	32 (all A all B (relation(B) -> relation_dom_restriction(B,A) = relation_composition(identity_relation(A),B))) # label(t94_relat_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	33 (all A all B (in(A,B) -> -in(B,A))) # label(antisymmetry_r2_hidden) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	34 (all A all B (element(B,powerset(A)) -> element(subset_complement(A,B),powerset(A)))) # label(dt_k3_subset_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	35 (all A all B (relation(B) -> relation_dom(relation_dom_restriction(B,A)) = set_intersection2(relation_dom(B),A))) # label(t90_relat_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	36 (all A (relation(identity_relation(A)) & function(identity_relation(A)))) # label(fc2_funct_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	37 (all A (relation(A) -> (all B all C (C = fiber(A,B) <-> (all D (in(D,C) <-> D != B & in(ordered_pair(D,B),A))))))) # label(d1_wellord1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	38 (exists A -empty(A)) # label(rc2_xboole_0) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	39 (all A (relation(A) -> (well_founded_relation(A) <-> is_well_founded_in(A,relation_field(A))))) # label(t5_wellord1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	40 (exists A (epsilon_connected(A) & ordinal(A) & epsilon_transitive(A) & -empty(A))) # label(rc3_ordinal1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	41 (all A all B (relation(B) & relation(A) -> relation(set_difference(A,B)))) # label(fc3_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	42 (all A all B (relation(B) -> subset(relation_inverse_image(B,A),relation_dom(B)))) # label(t167_relat_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	43 (all A all B all C (unordered_pair(A,B) = C <-> (all D (in(D,C) <-> B = D | A = D)))) # label(d2_tarski) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	44 (all A ((all B -((all C all D B != ordered_pair(C,D)) & in(B,A))) <-> relation(A))) # label(d1_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	45 (all A (relation(A) -> (all B ((all C all D (in(ordered_pair(D,C),A) & in(ordered_pair(C,D),A) & in(D,B) & in(C,B) -> C = D)) <-> is_antisymmetric_in(A,B))))) # label(d4_relat_2) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	46 (all A all B (element(B,powerset(powerset(A))) -> element(union_of_subsets(A,B),powerset(A)))) # label(dt_k5_setfam_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	47 (all A all B (relation(B) -> subset(relation_image(B,A),relation_rng(B)))) # label(t144_relat_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	48 (all A all B all C (relation(C) -> (in(A,relation_field(relation_restriction(C,B))) -> in(A,relation_field(C)) & in(A,B)))) # label(t19_wellord1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	49 (all A all B (element(A,powerset(B)) <-> subset(A,B))) # label(t3_subset) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	50 (all A all B -((all C -(in(C,B) & (all D -(in(D,B) & in(D,C))))) & in(A,B))) # label(t7_tarski) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	51 (all A all B all C (element(B,powerset(A)) & element(C,powerset(A)) -> element(subset_difference(A,B,C),powerset(A)))) # label(dt_k6_subset_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	52 $T # label(dt_k1_zfmisc_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	53 (all A (relation(A) -> (well_ordering(A) <-> reflexive(A) & transitive(A) & well_founded_relation(A) & connected(A) & antisymmetric(A)))) # label(d4_wellord1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	54 (all A exists B (empty(B) & element(B,powerset(A)))) # label(rc2_subset_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	55 (all A empty_set != singleton(A)) # label(l1_zfmisc_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	56 (all A all B (-empty(A) -> -empty(set_union2(B,A)))) # label(fc3_xboole_0) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	57 (all A all B (in(A,B) -> subset(A,union(B)))) # label(t92_zfmisc_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	58 $T # label(dt_k2_xboole_0) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	59 (all A (relation(A) -> (well_founded_relation(A) <-> (all B -((all C -(in(C,B) & disjoint(fiber(A,C),B))) & B != empty_set & subset(B,relation_field(A))))))) # label(d2_wellord1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	60 (all A all B (in(A,B) -> B = set_union2(singleton(A),B))) # label(l23_zfmisc_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	61 (all A all B (subset(A,B) -> B = set_union2(A,set_difference(B,A)))) # label(t45_xboole_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	62 (all A all B (function(B) & relation(B) -> (subset(A,relation_rng(B)) -> relation_image(B,relation_inverse_image(B,A)) = A))) # label(t147_funct_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	63 (all A all B (relation(B) & function(B) -> (all C (function(C) & relation(C) -> (in(A,relation_dom(relation_composition(C,B))) <-> in(apply(C,A),relation_dom(B)) & in(A,relation_dom(C))))))) # label(t21_funct_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	64 (all A all B (in(A,B) -> element(A,B))) # label(t1_subset) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	65 (all A all B all C all D (in(B,D) & in(A,C) <-> in(ordered_pair(A,B),cartesian_product2(C,D)))) # label(l55_zfmisc_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	66 (all A (A != empty_set -> (all B (element(B,powerset(A)) -> (all C (element(C,A) -> (-in(C,B) -> in(C,subset_complement(A,B))))))))) # label(t50_subset_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	67 (all A all B (relation(B) -> relation_image(B,set_intersection2(relation_dom(B),A)) = relation_image(B,A))) # label(t145_relat_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	68 (all A all B set_intersection2(B,A) = set_intersection2(A,B)) # label(commutativity_k3_xboole_0) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	69 $T # label(dt_k1_setfam_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	70 (all A (relation(A) -> (is_transitive_in(A,relation_field(A)) <-> transitive(A)))) # label(d16_relat_2) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	71 (all A (epsilon_connected(A) <-> (all B all C -(in(B,A) & B != C & -in(C,B) & -in(B,C) & in(C,A))))) # label(d3_ordinal1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	72 $T # label(dt_k3_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	73 (all A (relation(A) -> (all B all C (C = relation_inverse_image(A,B) <-> (all D ((exists E (in(E,B) & in(ordered_pair(D,E),A))) <-> in(D,C))))))) # label(d14_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	74 (all A all B all C (in(A,C) & in(B,C) <-> subset(unordered_pair(A,B),C))) # label(t38_zfmisc_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	75 (all A (ordinal(A) -> epsilon_connected(A) & epsilon_transitive(A))) # label(cc1_ordinal1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	76 (all A empty_set = set_intersection2(A,empty_set)) # label(t2_boole) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	77 (all A all B (in(A,B) <-> subset(singleton(A),B))) # label(t37_zfmisc_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	78 (all A (relation(A) -> (all B (relation(B) -> ((all C all D (in(ordered_pair(C,D),B) <-> in(ordered_pair(C,D),A))) <-> A = B))))) # label(d2_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	79 (all A all B set_union2(A,A) = A) # label(idempotence_k2_xboole_0) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	80 (all A all B (relation(B) -> (all C (relation(C) -> ((all D all E (in(E,A) & in(ordered_pair(D,E),B) <-> in(ordered_pair(D,E),C))) <-> relation_rng_restriction(A,B) = C))))) # label(d12_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	81 (all A set_difference(A,empty_set) = A) # label(t3_boole) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	82 (all A (relation(A) -> (all B ((all C (in(C,B) -> in(ordered_pair(C,C),A))) <-> is_reflexive_in(A,B))))) # label(d1_relat_2) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	83 (exists A (relation(A) & relation_empty_yielding(A) & function(A))) # label(rc4_funct_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	84 (all A all B all C (function(C) & relation(C) -> (in(ordered_pair(A,B),C) <-> in(A,relation_dom(C)) & B = apply(C,A)))) # label(t8_funct_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	85 (all A all B (relation(B) & empty(A) -> empty(relation_composition(A,B)) & relation(relation_composition(A,B)))) # label(fc9_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	86 (all A (relation(A) & function(A) -> (one_to_one(A) <-> (all B all C (in(C,relation_dom(A)) & apply(A,C) = apply(A,B) & in(B,relation_dom(A)) -> B = C))))) # label(d8_funct_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	87 (all A all B (relation(B) -> relation_rng_restriction(A,relation_dom_restriction(B,A)) = relation_restriction(B,A))) # label(t18_wellord1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	88 (all A all B ordered_pair(A,B) = unordered_pair(unordered_pair(A,B),singleton(A))) # label(d5_tarski) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	89 (all A all B (relation(B) -> subset(relation_rng_restriction(A,B),B))) # label(t117_relat_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	90 (exists A (-empty(A) & relation(A))) # label(rc2_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	91 (all A all B all C (relation_of2_as_subset(C,A,B) -> subset(relation_rng(C),B) & subset(relation_dom(C),A))) # label(t12_relset_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	92 $T # label(dt_k1_ordinal1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	93 $T # label(dt_k10_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	94 (all A all B all C (subset(A,B) -> subset(cartesian_product2(A,C),cartesian_product2(B,C)) & subset(cartesian_product2(C,A),cartesian_product2(C,B)))) # label(t118_zfmisc_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	95 (all A all B all C -(element(B,powerset(C)) & empty(C) & in(A,B))) # label(t5_subset) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	96 (all A all B set_difference(A,B) = set_difference(set_union2(A,B),B)) # label(t40_xboole_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	97 (all A (relation(A) -> (all B (relation(B) -> (subset(relation_rng(A),relation_dom(B)) -> relation_dom(A) = relation_dom(relation_composition(A,B))))))) # label(t46_relat_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	98 (all A (relation(A) -> relation(relation_inverse(A)))) # label(dt_k4_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	99 (all A (empty(A) -> empty(relation_inverse(A)) & relation(relation_inverse(A)))) # label(fc11_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	100 (all A (epsilon_transitive(A) -> (all B (ordinal(B) -> (proper_subset(A,B) -> in(A,B)))))) # label(t21_ordinal1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	101 $T # label(dt_k1_tarski) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	102 (all A all B all C (subset(C,cartesian_product2(A,B)) <-> relation_of2(C,A,B))) # label(d1_relset_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	103 (all A all B (ordinal(B) -> (in(A,B) -> ordinal(A)))) # label(t23_ordinal1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	104 (all A (relation(A) -> (all B (relation(B) -> (all C (relation(C) & function(C) -> (relation_isomorphism(A,B,C) -> relation_isomorphism(B,A,function_inverse(C))))))))) # label(t49_wellord1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	105 (all A (-empty(A) & relation(A) -> -empty(relation_dom(A)))) # label(fc5_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	106 (all A all B -proper_subset(A,A)) # label(irreflexivity_r2_xboole_0) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	107 (all A all B all C all D (relation(D) -> (in(ordered_pair(A,B),D) & in(A,C) <-> in(ordered_pair(A,B),relation_composition(identity_relation(C),D))))) # label(t74_relat_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	108 (all A all B (element(B,powerset(powerset(A))) -> (empty_set != B -> subset_difference(A,cast_to_subset(A),meet_of_subsets(A,B)) = union_of_subsets(A,complements_of_subsets(A,B))))) # label(t48_setfam_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	109 (all A (ordinal(A) -> epsilon_transitive(union(A)) & epsilon_connected(union(A)) & ordinal(union(A)))) # label(fc4_ordinal1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	110 (all A all B (relation(B) & relation(A) -> relation(set_intersection2(A,B)))) # label(fc1_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	111 (all A (empty(A) -> empty_set = A)) # label(t6_boole) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	112 (all A all B all C all D ((all E (-(E != B & E != C & A != E) <-> in(E,D))) <-> unordered_triple(A,B,C) = D)) # label(d1_enumset1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	113 (all A all B all C (relation(C) -> (in(ordered_pair(A,B),C) -> in(A,relation_dom(C)) & in(B,relation_rng(C))))) # label(t20_relat_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	114 (all A relation(identity_relation(A))) # label(dt_k6_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	115 $T # label(dt_k1_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	116 (all A all B all C (relation(C) -> ((exists D (in(ordered_pair(A,D),C) & in(D,B) & in(D,relation_rng(C)))) <-> in(A,relation_inverse_image(C,B))))) # label(t166_relat_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	117 (exists A (one_to_one(A) & function(A) & relation(A))) # label(rc3_funct_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	118 (all A (relation(A) -> (antisymmetric(A) <-> (all B all C (in(ordered_pair(C,B),A) & in(ordered_pair(B,C),A) -> C = B))))) # label(l3_wellord1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	119 (all A all B (element(B,powerset(powerset(A))) -> union_of_subsets(A,B) = union(B))) # label(redefinition_k5_setfam_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	120 (all A all B (element(B,powerset(A)) -> subset_complement(A,B) = set_difference(A,B))) # label(d5_subset_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	121 (all A (relation(A) -> (all B (is_well_founded_in(A,B) <-> (all C -(empty_set != C & (all D -(in(D,C) & disjoint(fiber(A,D),C))) & subset(C,B))))))) # label(d3_wellord1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	122 (all A all B (in(B,A) -> apply(identity_relation(A),B) = B)) # label(t35_funct_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	123 (all A all B (relation(B) -> (reflexive(B) -> reflexive(relation_restriction(B,A))))) # label(t22_wellord1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	124 (all A all B (ordinal(B) & ordinal(A) -> (ordinal_subset(A,B) <-> subset(A,B)))) # label(redefinition_r1_ordinal1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	125 (all A (being_limit_ordinal(A) <-> A = union(A))) # label(d6_ordinal1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	126 (all A all B (disjoint(A,B) -> disjoint(B,A))) # label(symmetry_r1_xboole_0) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	127 (all A all B (subset(A,singleton(B)) <-> singleton(B) = A | A = empty_set)) # label(l4_zfmisc_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	128 (all A all B (ordinal(B) -> -(subset(A,B) & A != empty_set & (all C (ordinal(C) -> -((all D (ordinal(D) -> (in(D,A) -> ordinal_subset(C,D)))) & in(C,A))))))) # label(t32_ordinal1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	129 (all A all B (relation(B) -> subset(relation_dom(relation_rng_restriction(A,B)),relation_dom(B)))) # label(l29_wellord1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	130 (all A (relation(A) & function(A) -> function(function_inverse(A)) & relation(function_inverse(A)))) # label(dt_k2_funct_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	131 $T # label(dt_k2_zfmisc_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	132 (all A all B all C (C = set_union2(A,B) <-> (all D (in(D,C) <-> in(D,B) | in(D,A))))) # label(d2_xboole_0) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	133 (all A all B all C (relation_of2_as_subset(C,A,B) <-> relation_of2(C,A,B))) # label(redefinition_m2_relset_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	134 (all A all B -(empty(B) & B != A & empty(A))) # label(t8_boole) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	135 (all A -empty(singleton(A))) # label(fc2_subset_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.27	136 (all A (relation(A) -> (reflexive(A) <-> (all B (in(B,relation_field(A)) -> in(ordered_pair(B,B),A)))))) # label(l1_wellord1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	137 (all A (ordinal(A) -> ((all B (ordinal(B) -> (in(B,A) -> in(succ(B),A)))) <-> being_limit_ordinal(A)))) # label(t41_ordinal1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.27	138 (all A (relation(A) -> (all B (relation(B) -> (all C (function(C) & relation(C) -> (relation_isomorphism(A,B,C) -> (reflexive(A) -> reflexive(B)) & (transitive(A) -> transitive(B)) & (connected(A) -> connected(B)) & (well_founded_relation(A) -> well_founded_relation(B)) & (antisymmetric(A) -> antisymmetric(B))))))))) # label(t53_wellord1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	139 (all A (relation(A) -> ((all B all C -in(ordered_pair(B,C),A)) -> empty_set = A))) # label(t56_relat_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	140 (all A set_union2(A,empty_set) = A) # label(t1_boole) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	141 (all A all B all C (singleton(A) = unordered_pair(B,C) -> B = C)) # label(t9_zfmisc_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	142 (all A all B (relation_empty_yielding(A) & relation(A) -> relation_empty_yielding(relation_dom_restriction(A,B)) & relation(relation_dom_restriction(A,B)))) # label(fc13_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	143 (all A all B (relation(B) -> (subset(A,relation_field(B)) & well_ordering(B) -> A = relation_field(relation_restriction(B,A))))) # label(t39_wellord1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	144 (all A (empty(A) & function(A) & relation(A) -> function(A) & one_to_one(A) & relation(A))) # label(cc2_funct_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	145 (all A all B (ordinal(B) & ordinal(A) -> ordinal_subset(A,B) | ordinal_subset(B,A))) # label(connectedness_r1_ordinal1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	146 $T # label(dt_k3_xboole_0) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	147 (all A ((all B (in(B,A) -> ordinal(B) & subset(B,A))) -> ordinal(A))) # label(t31_ordinal1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	148 (all A all B (subset(A,B) -> A = set_intersection2(A,B))) # label(t28_xboole_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	149 (all A (relation(A) -> (all B ((all C all D -(C != D & -in(ordered_pair(C,D),A) & -in(ordered_pair(D,C),A) & in(D,B) & in(C,B))) <-> is_connected_in(A,B))))) # label(d6_relat_2) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	150 (all A in(A,succ(A))) # label(t10_ordinal1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	151 (all A (epsilon_transitive(A) & epsilon_connected(A) -> ordinal(A))) # label(cc2_ordinal1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	152 (all A all B (relation(B) -> subset(relation_rng(relation_dom_restriction(B,A)),relation_rng(B)))) # label(t99_relat_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	153 (all A (function(A) & relation(A) -> (one_to_one(A) -> one_to_one(function_inverse(A))))) # label(t62_funct_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	154 (all A (relation(A) -> (well_ordering(A) <-> well_orders(A,relation_field(A))))) # label(t8_wellord1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	155 (all A all B subset(A,set_union2(A,B))) # label(t7_xboole_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	156 (all A (relation(A) -> (all B ((all C all D all E (in(ordered_pair(D,E),A) & in(ordered_pair(C,D),A) & in(E,B) & in(D,B) & in(C,B) -> in(ordered_pair(C,E),A))) <-> is_transitive_in(A,B))))) # label(d8_relat_2) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	157 (all A (relation(A) -> (relation_dom(A) = empty_set | empty_set = relation_rng(A) -> empty_set = A))) # label(t64_relat_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	158 (all A all B (element(B,powerset(powerset(A))) -> (empty_set != B -> subset_difference(A,cast_to_subset(A),union_of_subsets(A,B)) = meet_of_subsets(A,complements_of_subsets(A,B))))) # label(t47_setfam_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	159 (all A (ordinal(A) <-> epsilon_transitive(A) & epsilon_connected(A))) # label(d4_ordinal1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	160 (exists A (empty(A) & function(A) & relation(A))) # label(rc2_funct_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	161 (all A all B all C all D (ordered_pair(A,B) = ordered_pair(C,D) -> B = D & A = C)) # label(t33_zfmisc_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	162 (all A (relation(A) & function(A) -> (one_to_one(A) -> relation_inverse(A) = function_inverse(A)))) # label(d9_funct_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	163 (all A all B (proper_subset(A,B) -> -proper_subset(B,A))) # label(antisymmetry_r2_xboole_0) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	164 (all A all B all C (C = cartesian_product2(A,B) <-> (all D ((exists E exists F (in(F,B) & ordered_pair(E,F) = D & in(E,A))) <-> in(D,C))))) # label(d2_zfmisc_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	165 (all A all B -(proper_subset(B,A) & subset(A,B))) # label(t60_xboole_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	166 (all A all B (ordinal(A) & ordinal(B) -> ordinal_subset(A,A))) # label(reflexivity_r1_ordinal1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	167 (all A all B all C (element(C,powerset(A)) -> -(in(B,C) & in(B,subset_complement(A,C))))) # label(t54_subset_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	168 (all A (relation(A) -> ((all B all C -(in(B,relation_field(A)) & B != C & -in(ordered_pair(B,C),A) & -in(ordered_pair(C,B),A) & in(C,relation_field(A)))) <-> connected(A)))) # label(l4_wellord1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	169 (all A (function(A) & one_to_one(A) & relation(A) -> function(relation_inverse(A)) & relation(relation_inverse(A)))) # label(fc3_funct_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	170 (all A all B all C all D -(unordered_pair(A,B) = unordered_pair(C,D) & A != D & A != C)) # label(t10_zfmisc_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	171 (all A all B (element(B,powerset(A)) -> (all C (element(C,powerset(A)) -> (subset(B,subset_complement(A,C)) <-> disjoint(B,C)))))) # label(t43_subset_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	172 (all A all B exists C relation_of2(C,A,B)) # label(existence_m1_relset_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	173 $T # label(dt_k4_tarski) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	174 (all A all B all C (element(B,powerset(A)) & element(C,powerset(A)) -> subset_difference(A,B,C) = set_difference(B,C))) # label(redefinition_k6_subset_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	175 $T # label(dt_k1_xboole_0) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	176 (all A (ordinal(A) -> (all B (ordinal(B) -> (in(A,B) <-> ordinal_subset(succ(A),B)))))) # label(t33_ordinal1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	177 (all A (relation(A) -> (reflexive(A) <-> is_reflexive_in(A,relation_field(A))))) # label(d9_relat_2) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	178 (all A subset(empty_set,A)) # label(t2_xboole_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	179 (all A all B (relation(B) -> relation(relation_rng_restriction(A,B)))) # label(dt_k8_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	180 (all A (relation(A) & function(A) -> (one_to_one(A) -> relation_dom(function_inverse(A)) = relation_rng(A) & relation_rng(function_inverse(A)) = relation_dom(A)))) # label(t55_funct_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	181 (all A all B all C (set_intersection2(A,B) = C <-> (all D (in(D,C) <-> in(D,A) & in(D,B))))) # label(d3_xboole_0) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	182 (all A all B all C (relation(C) -> (subset(A,B) -> subset(relation_inverse_image(C,A),relation_inverse_image(C,B))))) # label(t178_relat_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	183 (all A all B (relation(B) -> relation_rng(relation_rng_restriction(A,B)) = set_intersection2(relation_rng(B),A))) # label(t119_relat_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	184 (all A (relation(A) -> (all B (relation(B) -> (all C (function(C) & relation(C) -> (well_ordering(A) & relation_isomorphism(A,B,C) -> well_ordering(B)))))))) # label(t54_wellord1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	185 (all A succ(A) = set_union2(A,singleton(A))) # label(d1_ordinal1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	186 (all A all B (function(B) & relation(B) -> (B = identity_relation(A) <-> A = relation_dom(B) & (all C (in(C,A) -> C = apply(B,C)))))) # label(t34_funct_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	187 $T # label(dt_k2_tarski) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	188 (all A all B (relation(B) -> (well_founded_relation(B) -> well_founded_relation(relation_restriction(B,A))))) # label(t31_wellord1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	189 (exists A (relation(A) & function(A))) # label(rc1_funct_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	190 (all A all B (-in(A,B) -> disjoint(singleton(A),B))) # label(l28_zfmisc_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	191 (all A all B (relation(B) -> subset(relation_rng(relation_rng_restriction(A,B)),A))) # label(t116_relat_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	192 (all A all B all C (element(C,powerset(cartesian_product2(A,B))) -> relation(C))) # label(cc1_relset_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	193 (exists A (relation(A) & one_to_one(A) & epsilon_connected(A) & ordinal(A) & epsilon_transitive(A) & empty(A) & function(A))) # label(rc2_ordinal1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	194 $T # label(dt_k9_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	195 (all A all B (A = empty_set | A = singleton(B) <-> subset(A,singleton(B)))) # label(t39_zfmisc_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	196 (all A all B A = set_intersection2(A,A)) # label(idempotence_k3_xboole_0) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	197 (all A all B (disjoint(A,B) <-> set_intersection2(A,B) = empty_set)) # label(d7_xboole_0) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	198 (all A all B (subset(singleton(A),singleton(B)) -> B = A)) # label(t6_zfmisc_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	199 (all A all B (element(B,powerset(powerset(A))) -> (all C (element(C,powerset(powerset(A))) -> (C = complements_of_subsets(A,B) <-> (all D (element(D,powerset(A)) -> (in(subset_complement(A,D),B) <-> in(D,C))))))))) # label(d8_setfam_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	200 (all A all B set_intersection2(A,B) = set_difference(A,set_difference(A,B))) # label(t48_xboole_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	201 (all A all B -(in(A,B) & empty(B))) # label(t7_boole) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	202 (all A all B (in(A,B) -> subset(A,union(B)))) # label(l50_zfmisc_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	203 (all A all B (-empty(B) & -empty(A) -> -empty(cartesian_product2(A,B)))) # label(fc4_subset_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	204 (all A all B (subset(A,B) <-> (all C (in(C,A) -> in(C,B))))) # label(d3_tarski) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	205 (all A all B (function(B) & relation(B) -> (all C (relation(C) & function(C) -> (in(A,relation_dom(relation_composition(C,B))) -> apply(relation_composition(C,B),A) = apply(B,apply(C,A))))))) # label(t22_funct_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	206 (all A all B (subset(A,B) -> set_union2(A,B) = B)) # label(t12_xboole_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	207 (all A all B ((A = empty_set -> (B = set_meet(A) <-> B = empty_set)) & (A != empty_set -> (set_meet(A) = B <-> (all C (in(C,B) <-> (all D (in(D,A) -> in(C,D))))))))) # label(d1_setfam_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	208 (all A all B (relation(B) -> (well_ordering(B) -> well_ordering(relation_restriction(B,A))))) # label(t32_wellord1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	209 (all A (relation(A) -> (all B ((all C ((exists D in(ordered_pair(D,C),A)) <-> in(C,B))) <-> B = relation_rng(A))))) # label(d5_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	210 (all A (empty(A) -> epsilon_connected(A) & ordinal(A) & epsilon_transitive(A))) # label(cc3_ordinal1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	211 (all A all B (relation(B) -> subset(relation_rng(relation_rng_restriction(A,B)),relation_rng(B)))) # label(t118_relat_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	212 (all A all B all C (relation(C) -> (in(ordered_pair(A,B),C) -> in(B,relation_field(C)) & in(A,relation_field(C))))) # label(t30_relat_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	213 (all A all B (element(B,powerset(powerset(A))) -> -(empty_set = complements_of_subsets(A,B) & B != empty_set))) # label(t46_setfam_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	214 (all A all B (relation(B) -> (subset(A,relation_dom(B)) -> subset(A,relation_inverse_image(B,relation_image(B,A)))))) # label(t146_funct_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	215 (all A all B ((-empty(A) -> (element(B,A) <-> in(B,A))) & (empty(A) -> (empty(B) <-> element(B,A))))) # label(d2_subset_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	216 (exists A (relation(A) & relation_empty_yielding(A))) # label(rc3_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	217 (all A all B (-empty(A) -> -empty(set_union2(A,B)))) # label(fc2_xboole_0) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	218 (all A all B subset(set_difference(A,B),A)) # label(t36_xboole_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	219 (all A (relation(A) -> subset(A,cartesian_product2(relation_dom(A),relation_rng(A))))) # label(t21_relat_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	220 (exists A empty(A)) # label(rc1_xboole_0) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	221 $T # label(dt_m1_relset_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	222 (all A all B subset(set_intersection2(A,B),A)) # label(t17_xboole_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	223 (all A all B (relation(B) -> (B = identity_relation(A) <-> (all C all D (in(C,A) & C = D <-> in(ordered_pair(C,D),B)))))) # label(d10_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	224 (all A (relation(A) -> (all B (B = relation_dom(A) <-> (all C (in(C,B) <-> (exists D in(ordered_pair(C,D),A)))))))) # label(d4_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	225 (all A all B (element(B,powerset(powerset(A))) -> B = complements_of_subsets(A,complements_of_subsets(A,B)))) # label(involutiveness_k7_setfam_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	226 (all A all B (element(B,powerset(A)) -> B = subset_complement(A,subset_complement(A,B)))) # label(involutiveness_k3_subset_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	227 (all A all B all C (relation(C) -> (in(A,relation_rng(relation_rng_restriction(B,C))) <-> in(A,B) & in(A,relation_rng(C))))) # label(t115_relat_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	228 (all A (relation(A) -> (all B relation_restriction(A,B) = set_intersection2(A,cartesian_product2(B,B))))) # label(d6_wellord1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	229 (all A all B all C (set_difference(A,B) = C <-> (all D (-in(D,B) & in(D,A) <-> in(D,C))))) # label(d4_xboole_0) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	230 $T # label(dt_k1_funct_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	231 (all A all B all C (element(B,powerset(C)) & in(A,B) -> element(A,C))) # label(t4_subset) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	232 (all A all B all C (relation(C) & function(C) -> (in(B,relation_dom(relation_dom_restriction(C,A))) -> apply(C,B) = apply(relation_dom_restriction(C,A),B)))) # label(t70_funct_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	233 $T # label(dt_k1_wellord1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	234 (all A all B (-(disjoint(A,B) & (exists C (in(C,A) & in(C,B)))) & -((all C -(in(C,B) & in(C,A))) & -disjoint(A,B)))) # label(t3_xboole_0) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	235 (all A (relation(A) -> (all B all C (relation(C) -> ((all D all E (in(D,B) & in(ordered_pair(D,E),A) <-> in(ordered_pair(D,E),C))) <-> relation_dom_restriction(A,B) = C))))) # label(d11_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	236 (all A (relation(A) -> (all B (relation(B) -> (subset(relation_dom(A),relation_rng(B)) -> relation_rng(relation_composition(B,A)) = relation_rng(A)))))) # label(t47_relat_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	237 (all A (subset(A,empty_set) -> empty_set = A)) # label(t3_xboole_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	238 (all A all B (relation(B) & function(B) -> relation(relation_rng_restriction(A,B)) & function(relation_rng_restriction(A,B)))) # label(fc5_funct_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	239 (all A all B (subset(A,B) <-> empty_set = set_difference(A,B))) # label(l32_xboole_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	240 (all A all B (relation(B) -> (transitive(B) -> transitive(relation_restriction(B,A))))) # label(t24_wellord1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	241 (all A all B ((all C (in(C,A) <-> in(C,B))) -> B = A)) # label(t2_tarski) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	242 (all A all B all C (subset(A,B) & subset(B,C) -> subset(A,C))) # label(t1_xboole_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	243 (all A all B ((all C (in(C,B) <-> C = A)) <-> B = singleton(A))) # label(d1_tarski) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	244 (all A (relation(A) -> (all B (relation(B) -> ((all C all D (in(ordered_pair(C,D),A) -> in(ordered_pair(C,D),B))) <-> subset(A,B)))))) # label(d3_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	245 (all A all B (A = set_difference(A,singleton(B)) <-> -in(B,A))) # label(t65_zfmisc_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	246 (all A all B (relation(B) -> -(subset(A,relation_rng(B)) & empty_set = relation_inverse_image(B,A) & A != empty_set))) # label(t174_relat_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	247 (all A (relation(A) -> (all B (relation(B) -> (subset(A,B) -> subset(relation_dom(A),relation_dom(B)) & subset(relation_rng(A),relation_rng(B))))))) # label(t25_relat_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	248 (all A all B ((all C ((exists D (in(C,D) & in(D,A))) <-> in(C,B))) <-> union(A) = B)) # label(d4_tarski) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	249 (all A all B exists C relation_of2_as_subset(C,A,B)) # label(existence_m2_relset_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	250 (all A -empty(powerset(A))) # label(fc1_subset_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	251 (all A all B -(in(A,B) & disjoint(singleton(A),B))) # label(l25_zfmisc_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	252 (all A all B (relation(B) -> subset(relation_dom_restriction(B,A),B))) # label(t88_relat_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	253 (all A all B set_union2(A,set_difference(B,A)) = set_union2(A,B)) # label(t39_xboole_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	254 (all A (ordinal(A) -> -empty(succ(A)) & epsilon_connected(succ(A)) & ordinal(succ(A)) & epsilon_transitive(succ(A)))) # label(fc3_ordinal1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	255 (all A all B all C (subset(A,B) -> subset(set_intersection2(A,C),set_intersection2(B,C)))) # label(t26_xboole_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	256 $T # label(dt_k4_xboole_0) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	257 (all A all B all C (relation(C) & function(C) -> (in(B,A) & in(B,relation_dom(C)) <-> in(B,relation_dom(relation_dom_restriction(C,A)))))) # label(l82_funct_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	258 $T # label(dt_k2_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	259 (all A (empty(A) -> empty(relation_dom(A)) & relation(relation_dom(A)))) # label(fc7_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	260 (all A (relation(A) -> (all B (relation(B) -> (all C (function(C) & relation(C) -> ((all D all E (in(ordered_pair(D,E),A) <-> in(ordered_pair(apply(C,D),apply(C,E)),B) & in(E,relation_field(A)) & in(D,relation_field(A)))) & one_to_one(C) & relation_field(B) = relation_rng(C) & relation_field(A) = relation_dom(C) <-> relation_isomorphism(A,B,C)))))))) # label(d7_wellord1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.28	261 (all A all B all C (relation(C) -> subset(fiber(relation_restriction(C,A),B),fiber(C,B)))) # label(t21_wellord1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.28	262 (all A (relation(A) -> (all B (well_orders(A,B) <-> is_antisymmetric_in(A,B) & is_connected_in(A,B) & is_well_founded_in(A,B) & is_transitive_in(A,B) & is_reflexive_in(A,B))))) # label(d5_wellord1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.29	263 (all A all B (powerset(A) = B <-> (all C (subset(C,A) <-> in(C,B))))) # label(d1_zfmisc_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.29	264 (all A all B subset(A,A)) # label(reflexivity_r1_tarski) # label(axiom) # label(non_clause).  [assumption].
1.00/1.29	265 (all A (empty(A) -> relation(A))) # label(cc1_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.29	266 (all A (relation(A) -> relation_rng(A) = relation_image(A,relation_dom(A)))) # label(t146_relat_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.29	267 (all A all B (relation(A) & function(A) & function(B) & relation(B) -> function(relation_composition(A,B)) & relation(relation_composition(A,B)))) # label(fc1_funct_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.29	268 (all A all B all C all D (relation_of2_as_subset(D,C,A) -> (subset(relation_rng(D),B) -> relation_of2_as_subset(D,C,B)))) # label(t14_relset_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.29	269 (all A (relation(A) -> (all B (relation(B) -> subset(relation_rng(relation_composition(A,B)),relation_rng(B)))))) # label(t45_relat_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.29	270 (all A (relation(A) -> relation_field(A) = set_union2(relation_dom(A),relation_rng(A)))) # label(d6_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.29	271 (all A all B (relation(A) & relation(B) -> relation(relation_composition(A,B)))) # label(dt_k5_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.29	272 (all A all B all C (relation_of2_as_subset(C,A,B) -> element(C,powerset(cartesian_product2(A,B))))) # label(dt_m2_relset_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.29	273 (all A (relation(A) -> relation_dom(A) = relation_rng(relation_inverse(A)) & relation_dom(relation_inverse(A)) = relation_rng(A))) # label(t37_relat_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.29	274 (all A (relation(A) -> (is_antisymmetric_in(A,relation_field(A)) <-> antisymmetric(A)))) # label(d12_relat_2) # label(axiom) # label(non_clause).  [assumption].
1.00/1.29	275 (all A exists B (in(A,B) & (all C all D (subset(D,C) & in(C,B) -> in(D,B))) & (all C (in(C,B) -> in(powerset(C),B))) & (all C -(subset(C,B) & -in(C,B) & -are_equipotent(C,B))))) # label(t136_zfmisc_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.29	276 $T # label(dt_k3_tarski) # label(axiom) # label(non_clause).  [assumption].
1.00/1.29	277 (all A (relation(A) & function(A) -> (one_to_one(A) -> (all B (function(B) & relation(B) -> (B = function_inverse(A) <-> relation_rng(A) = relation_dom(B) & (all C all D ((D = apply(B,C) & in(C,relation_rng(A)) -> C = apply(A,D) & in(D,relation_dom(A))) & (apply(A,D) = C & in(D,relation_dom(A)) -> D = apply(B,C) & in(C,relation_rng(A))))))))))) # label(t54_funct_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.29	278 (all A (relation_dom(identity_relation(A)) = A & relation_rng(identity_relation(A)) = A)) # label(t71_relat_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.29	279 (all A (relation(A) -> A = relation_inverse(relation_inverse(A)))) # label(involutiveness_k4_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.29	280 (all A all B all C (relation(C) -> (in(A,relation_dom(relation_dom_restriction(C,B))) <-> in(A,relation_dom(C)) & in(A,B)))) # label(t86_relat_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.29	281 (all A all B all C (subset(A,B) -> in(C,A) | subset(A,set_difference(B,singleton(C))))) # label(l3_zfmisc_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.29	282 (all A all B (in(A,B) -> set_union2(singleton(A),B) = B)) # label(t46_zfmisc_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.29	283 (all A all B (set_difference(A,B) = A <-> disjoint(A,B))) # label(t83_xboole_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.29	284 (all A all B all C all D (subset(C,D) & subset(A,B) -> subset(cartesian_product2(A,C),cartesian_product2(B,D)))) # label(t119_zfmisc_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.29	285 (all A set_difference(empty_set,A) = empty_set) # label(t4_boole) # label(axiom) # label(non_clause).  [assumption].
1.00/1.29	286 (all A (relation(A) -> (empty_set = relation_rng(A) <-> empty_set = relation_dom(A)))) # label(t65_relat_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.29	287 (all A (relation(A) -> (all B all C ((all D ((exists E (in(E,B) & in(ordered_pair(E,D),A))) <-> in(D,C))) <-> C = relation_image(A,B))))) # label(d13_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.29	288 (all A all B (relation(B) & function(B) -> (in(A,relation_rng(B)) & one_to_one(B) -> apply(B,apply(function_inverse(B),A)) = A & apply(relation_composition(function_inverse(B),B),A) = A))) # label(t57_funct_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.29	289 (all A A = cast_to_subset(A)) # label(d4_subset_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.29	290 (all A all B (relation(B) -> subset(relation_field(relation_restriction(B,A)),A) & subset(relation_field(relation_restriction(B,A)),relation_field(B)))) # label(t20_wellord1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.29	291 (all A all B -empty(unordered_pair(A,B))) # label(fc3_subset_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.29	292 (all A all B set_union2(A,B) = set_union2(B,A)) # label(commutativity_k2_xboole_0) # label(axiom) # label(non_clause).  [assumption].
1.00/1.29	293 (all A all B (subset(singleton(A),B) <-> in(A,B))) # label(l2_zfmisc_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.29	294 (all A all B all C (relation(C) -> ((exists D (in(D,relation_dom(C)) & in(ordered_pair(D,A),C) & in(D,B))) <-> in(A,relation_image(C,B))))) # label(t143_relat_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.29	295 (all A (-empty(A) -> (exists B (element(B,powerset(A)) & -empty(B))))) # label(rc1_subset_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.29	296 (all A all B (element(B,powerset(powerset(A))) -> set_meet(B) = meet_of_subsets(A,B))) # label(redefinition_k6_setfam_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.29	297 (all A all B all C (relation(C) & function(C) -> (in(B,A) -> apply(C,B) = apply(relation_dom_restriction(C,A),B)))) # label(t72_funct_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.29	298 $T # label(dt_k1_enumset1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.29	299 (all A (relation(A) -> (all B (relation(B) -> relation_rng(relation_composition(A,B)) = relation_image(B,relation_rng(A)))))) # label(t160_relat_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.29	300 $T # label(dt_m1_subset_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.29	301 (all A all B (B != A & subset(A,B) <-> proper_subset(A,B))) # label(d8_xboole_0) # label(axiom) # label(non_clause).  [assumption].
1.00/1.29	302 (all A all B -empty(ordered_pair(A,B))) # label(fc1_zfmisc_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.29	303 (all A all B (element(B,powerset(A)) -> (all C (in(C,B) -> in(C,A))))) # label(l3_subset_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.29	304 (all A all B (relation(B) -> (connected(B) -> connected(relation_restriction(B,A))))) # label(t23_wellord1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.29	305 (all A all B all C (singleton(A) = unordered_pair(B,C) -> B = A)) # label(t8_zfmisc_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.29	306 (all A all B all C (subset(C,B) & subset(A,B) -> subset(set_union2(A,C),B))) # label(t8_xboole_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.29	307 (all A (ordinal(A) -> -(being_limit_ordinal(A) & (exists B (ordinal(B) & A = succ(B)))) & -((all B (ordinal(B) -> A != succ(B))) & -being_limit_ordinal(A)))) # label(t42_ordinal1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.29	308 (all A (-empty(A) & relation(A) -> -empty(relation_rng(A)))) # label(fc6_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.29	309 (all A (empty(A) -> empty(relation_rng(A)) & relation(relation_rng(A)))) # label(fc8_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.29	310 (all A all B all C (relation(C) -> (in(A,relation_restriction(C,B)) <-> in(A,C) & in(A,cartesian_product2(B,B))))) # label(t16_wellord1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.29	311 (all A all B all C all D (in(ordered_pair(A,B),cartesian_product2(C,D)) <-> in(A,C) & in(B,D))) # label(t106_zfmisc_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.34	312 (all A (relation(A) -> (all B (relation(B) -> (all C (relation(C) -> ((all D all E (in(ordered_pair(D,E),C) <-> (exists F (in(ordered_pair(D,F),A) & in(ordered_pair(F,E),B))))) <-> C = relation_composition(A,B)))))))) # label(d8_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.34	313 (all A all B (relation(B) & function(B) -> (all C (function(C) & relation(C) -> (B = relation_dom_restriction(C,A) <-> relation_dom(B) = set_intersection2(relation_dom(C),A) & (all D (in(D,relation_dom(B)) -> apply(B,D) = apply(C,D)))))))) # label(t68_funct_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.34	314 (all A (relation(A) & function(A) -> (all B all C ((in(B,relation_dom(A)) -> (in(ordered_pair(B,C),A) <-> C = apply(A,B))) & (-in(B,relation_dom(A)) -> (empty_set = C <-> apply(A,B) = C)))))) # label(d4_funct_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.34	315 (all A (relation(A) -> (is_connected_in(A,relation_field(A)) <-> connected(A)))) # label(d14_relat_2) # label(axiom) # label(non_clause).  [assumption].
1.00/1.34	316 (all A (relation(A) & function(A) -> (all B all C (C = relation_inverse_image(A,B) <-> (all D (in(D,C) <-> in(apply(A,D),B) & in(D,relation_dom(A)))))))) # label(d13_funct_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.34	317 (all A all B (element(B,powerset(powerset(A))) -> element(complements_of_subsets(A,B),powerset(powerset(A))))) # label(dt_k7_setfam_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.34	318 (all A all B (relation(B) -> (antisymmetric(B) -> antisymmetric(relation_restriction(B,A))))) # label(t25_wellord1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.34	319 (all A all B all C (subset(A,B) -> subset(set_difference(A,C),set_difference(B,C)))) # label(t33_xboole_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.34	320 (all A unordered_pair(A,A) = singleton(A)) # label(t69_enumset1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.34	321 (all A all B (relation(A) & relation(B) -> relation(set_union2(A,B)))) # label(fc2_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.34	322 (all A all B (empty(A) & relation(B) -> relation(relation_composition(B,A)) & empty(relation_composition(B,A)))) # label(fc10_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.34	323 (all A all B (relation(B) & function(B) -> subset(relation_image(B,relation_inverse_image(B,A)),A))) # label(t145_funct_1) # label(lemma) # label(non_clause).  [assumption].
1.00/1.34	324 (exists A (empty(A) & relation(A))) # label(rc1_relat_1) # label(axiom) # label(non_clause).  [assumption].
1.00/1.34	325 -(all A all B all C all D (relation_of2_as_subset(D,C,A) -> (subset(A,B) -> relation_of2_as_subset(D,C,B)))) # label(t16_relset_1) # label(negated_conjecture) # label(non_clause).  [assumption].
1.00/1.34	
1.00/1.34	============================== end of process non-clausal formulas ===
1.00/1.34	
1.00/1.34	============================== PROCESS INITIAL CLAUSES ===============
1.00/1.34	
1.00/1.34	============================== PREDICATE ELIMINATION =================
1.00/1.34	326 -epsilon_transitive(A) | -in(B,A) | subset(B,A) # label(d2_ordinal1) # label(axiom).  [clausify(31)].
1.00/1.34	327 epsilon_transitive(c1) # label(rc1_ordinal1) # label(axiom).  [clausify(11)].
1.00/1.34	Derived: -in(A,c1) | subset(A,c1).  [resolve(326,a,327,a)].
1.00/1.34	328 epsilon_transitive(A) | in(f18(A),A) # label(d2_ordinal1) # label(axiom).  [clausify(31)].
1.00/1.34	Derived: in(f18(A),A) | -in(B,A) | subset(B,A).  [resolve(328,a,326,a)].
1.00/1.34	329 epsilon_transitive(A) | -subset(f18(A),A) # label(d2_ordinal1) # label(axiom).  [clausify(31)].
1.00/1.34	Derived: -subset(f18(A),A) | -in(B,A) | subset(B,A).  [resolve(329,a,326,a)].
1.00/1.34	330 epsilon_transitive(c3) # label(rc3_ordinal1) # label(axiom).  [clausify(40)].
1.00/1.34	Derived: -in(A,c3) | subset(A,c3).  [resolve(330,a,326,a)].
1.00/1.34	331 -ordinal(A) | epsilon_transitive(A) # label(cc1_ordinal1) # label(axiom).  [clausify(75)].
1.00/1.34	Derived: -ordinal(A) | -in(B,A) | subset(B,A).  [resolve(331,b,326,a)].
1.00/1.34	332 -epsilon_transitive(A) | -ordinal(B) | -proper_subset(A,B) | in(A,B) # label(t21_ordinal1) # label(lemma).  [clausify(100)].
1.00/1.41	Derived: -ordinal(A) | -proper_subset(c1,A) | in(c1,A).  [resolve(332,a,327,a)].
1.00/1.41	Derived: -ordinal(A) | -proper_subset(B,A) | in(B,A) | in(f18(B),B).  [resolve(332,a,328,a)].
1.00/1.41	Derived: -ordinal(A) | -proper_subset(B,A) | in(B,A) | -subset(f18(B),B).  [resolve(332,a,329,a)].
1.00/1.41	Derived: -ordinal(A) | -proper_subset(c3,A) | in(c3,A).  [resolve(332,a,330,a)].
1.00/1.41	Derived: -ordinal(A) | -proper_subset(B,A) | in(B,A) | -ordinal(B).  [resolve(332,a,331,b)].
1.00/1.41	333 -ordinal(A) | epsilon_transitive(union(A)) # label(fc4_ordinal1) # label(axiom).  [clausify(109)].
1.00/1.41	Derived: -ordinal(A) | -in(B,union(A)) | subset(B,union(A)).  [resolve(333,b,326,a)].
1.00/1.41	Derived: -ordinal(A) | -ordinal(B) | -proper_subset(union(A),B) | in(union(A),B).  [resolve(333,b,332,a)].
1.00/1.41	334 -epsilon_transitive(A) | -epsilon_connected(A) | ordinal(A) # label(cc2_ordinal1) # label(axiom).  [clausify(151)].
1.00/1.41	Derived: -epsilon_connected(c1) | ordinal(c1).  [resolve(334,a,327,a)].
1.00/1.41	Derived: -epsilon_connected(A) | ordinal(A) | in(f18(A),A).  [resolve(334,a,328,a)].
1.00/1.41	Derived: -epsilon_connected(A) | ordinal(A) | -subset(f18(A),A).  [resolve(334,a,329,a)].
1.00/1.41	Derived: -epsilon_connected(c3) | ordinal(c3).  [resolve(334,a,330,a)].
1.00/1.41	Derived: -epsilon_connected(union(A)) | ordinal(union(A)) | -ordinal(A).  [resolve(334,a,333,b)].
1.00/1.41	335 -ordinal(A) | epsilon_transitive(A) # label(d4_ordinal1) # label(axiom).  [clausify(159)].
1.00/1.41	336 ordinal(A) | -epsilon_transitive(A) | -epsilon_connected(A) # label(d4_ordinal1) # label(axiom).  [clausify(159)].
1.00/1.41	337 epsilon_transitive(empty_set) # label(fc2_ordinal1_AndRHS_AndLHS) # label(axiom).  [assumption].
1.00/1.41	Derived: -in(A,empty_set) | subset(A,empty_set).  [resolve(337,a,326,a)].
1.00/1.41	Derived: -ordinal(A) | -proper_subset(empty_set,A) | in(empty_set,A).  [resolve(337,a,332,a)].
1.00/1.41	338 epsilon_transitive(c9) # label(rc2_ordinal1) # label(axiom).  [clausify(193)].
1.00/1.41	Derived: -in(A,c9) | subset(A,c9).  [resolve(338,a,326,a)].
1.00/1.41	Derived: -ordinal(A) | -proper_subset(c9,A) | in(c9,A).  [resolve(338,a,332,a)].
1.00/1.41	Derived: -epsilon_connected(c9) | ordinal(c9).  [resolve(338,a,334,a)].
1.00/1.41	339 -empty(A) | epsilon_transitive(A) # label(cc3_ordinal1) # label(axiom).  [clausify(210)].
1.00/1.41	Derived: -empty(A) | -in(B,A) | subset(B,A).  [resolve(339,b,326,a)].
1.00/1.41	Derived: -empty(A) | -ordinal(B) | -proper_subset(A,B) | in(A,B).  [resolve(339,b,332,a)].
1.00/1.41	Derived: -empty(A) | -epsilon_connected(A) | ordinal(A).  [resolve(339,b,334,a)].
1.00/1.41	340 -ordinal(A) | epsilon_transitive(succ(A)) # label(fc3_ordinal1) # label(axiom).  [clausify(254)].
1.00/1.41	Derived: -ordinal(A) | -in(B,succ(A)) | subset(B,succ(A)).  [resolve(340,b,326,a)].
1.00/1.41	Derived: -ordinal(A) | -ordinal(B) | -proper_subset(succ(A),B) | in(succ(A),B).  [resolve(340,b,332,a)].
1.00/1.41	Derived: -ordinal(A) | -epsilon_connected(succ(A)) | ordinal(succ(A)).  [resolve(340,b,334,a)].
1.00/1.41	341 subset(A,cartesian_product2(B,C)) | -relation_of2(A,B,C) # label(d1_relset_1) # label(axiom).  [clausify(102)].
1.00/1.41	342 -subset(A,cartesian_product2(B,C)) | relation_of2(A,B,C) # label(d1_relset_1) # label(axiom).  [clausify(102)].
1.00/1.41	343 -relation_of2_as_subset(A,B,C) | relation_of2(A,B,C) # label(redefinition_m2_relset_1) # label(axiom).  [clausify(133)].
1.00/1.41	Derived: -relation_of2_as_subset(A,B,C) | subset(A,cartesian_product2(B,C)).  [resolve(343,b,341,b)].
1.00/1.41	344 relation_of2_as_subset(A,B,C) | -relation_of2(A,B,C) # label(redefinition_m2_relset_1) # label(axiom).  [clausify(133)].
1.00/1.41	Derived: relation_of2_as_subset(A,B,C) | -subset(A,cartesian_product2(B,C)).  [resolve(344,b,342,b)].
1.00/1.41	345 relation_of2(f67(A,B),A,B) # label(existence_m1_relset_1) # label(axiom).  [clausify(172)].
1.00/1.41	Derived: subset(f67(A,B),cartesian_product2(A,B)).  [resolve(345,a,341,b)].
1.00/1.41	Derived: relation_of2_as_subset(f67(A,B),A,B).  [resolve(345,a,344,b)].
1.00/1.41	346 being_limit_ordinal(A) | union(A) != A # label(d6_ordinal1) # label(axiom).  [clausify(125)].
1.00/1.41	347 -being_limit_ordinal(A) | union(A) = A # label(d6_ordinal1) # label(axiom).  [clausify(125)].
1.00/1.41	348 -ordinal(A) | ordinal(f51(A)) | being_limit_ordinal(A) # label(t41_ordinal1) # label(lemma).  [clausify(137)].
1.00/1.41	Derived: -ordinal(A) | ordinal(f51(A)) | union(A) = A.  [resolve(348,c,347,a)].
1.00/1.62	349 -ordinal(A) | in(f51(A),A) | being_limit_ordinal(A) # label(t41_ordinal1) # label(lemma).  [clausify(137)].
1.00/1.62	Derived: -ordinal(A) | in(f51(A),A) | union(A) = A.  [resolve(349,c,347,a)].
1.00/1.62	350 -ordinal(A) | -in(succ(f51(A)),A) | being_limit_ordinal(A) # label(t41_ordinal1) # label(lemma).  [clausify(137)].
1.00/1.62	Derived: -ordinal(A) | -in(succ(f51(A)),A) | union(A) = A.  [resolve(350,c,347,a)].
1.00/1.62	351 -ordinal(A) | -ordinal(B) | -in(B,A) | in(succ(B),A) | -being_limit_ordinal(A) # label(t41_ordinal1) # label(lemma).  [clausify(137)].
1.00/1.62	Derived: -ordinal(A) | -ordinal(B) | -in(B,A) | in(succ(B),A) | union(A) != A.  [resolve(351,e,346,a)].
1.00/1.62	Derived: -ordinal(A) | -ordinal(B) | -in(B,A) | in(succ(B),A) | -ordinal(A) | ordinal(f51(A)).  [resolve(351,e,348,c)].
1.00/1.62	Derived: -ordinal(A) | -ordinal(B) | -in(B,A) | in(succ(B),A) | -ordinal(A) | in(f51(A),A).  [resolve(351,e,349,c)].
1.00/1.62	Derived: -ordinal(A) | -ordinal(B) | -in(B,A) | in(succ(B),A) | -ordinal(A) | -in(succ(f51(A)),A).  [resolve(351,e,350,c)].
1.00/1.62	352 -ordinal(A) | -being_limit_ordinal(A) | -ordinal(B) | succ(B) != A # label(t42_ordinal1) # label(lemma).  [clausify(307)].
1.00/1.62	Derived: -ordinal(A) | -ordinal(B) | succ(B) != A | union(A) != A.  [resolve(352,b,346,a)].
1.00/1.62	Derived: -ordinal(A) | -ordinal(B) | succ(B) != A | -ordinal(A) | ordinal(f51(A)).  [resolve(352,b,348,c)].
1.00/1.62	Derived: -ordinal(A) | -ordinal(B) | succ(B) != A | -ordinal(A) | in(f51(A),A).  [resolve(352,b,349,c)].
1.00/1.62	Derived: -ordinal(A) | -ordinal(B) | succ(B) != A | -ordinal(A) | -in(succ(f51(A)),A).  [resolve(352,b,350,c)].
1.00/1.62	353 -ordinal(A) | ordinal(f106(A)) | being_limit_ordinal(A) # label(t42_ordinal1) # label(lemma).  [clausify(307)].
1.00/1.62	Derived: -ordinal(A) | ordinal(f106(A)) | union(A) = A.  [resolve(353,c,347,a)].
1.00/1.62	Derived: -ordinal(A) | ordinal(f106(A)) | -ordinal(A) | -ordinal(B) | -in(B,A) | in(succ(B),A).  [resolve(353,c,351,e)].
1.00/1.62	Derived: -ordinal(A) | ordinal(f106(A)) | -ordinal(A) | -ordinal(B) | succ(B) != A.  [resolve(353,c,352,b)].
1.00/1.62	354 -ordinal(A) | succ(f106(A)) = A | being_limit_ordinal(A) # label(t42_ordinal1) # label(lemma).  [clausify(307)].
1.00/1.62	Derived: -ordinal(A) | succ(f106(A)) = A | union(A) = A.  [resolve(354,c,347,a)].
1.00/1.62	Derived: -ordinal(A) | succ(f106(A)) = A | -ordinal(A) | -ordinal(B) | -in(B,A) | in(succ(B),A).  [resolve(354,c,351,e)].
1.00/1.62	Derived: -ordinal(A) | succ(f106(A)) = A | -ordinal(A) | -ordinal(B) | succ(B) != A.  [resolve(354,c,352,b)].
1.00/1.62	
1.00/1.62	============================== end predicate elimination =============
1.00/1.62	
1.00/1.62	Auto_denials:  (non-Horn, no changes).
1.00/1.62	
1.00/1.62	Term ordering decisions:
1.00/1.62	Function symbol KB weights:  empty_set=1. c1=1. c2=1. c3=1. c4=1. c5=1. c6=1. c7=1. c8=1. c9=1. c10=1. c11=1. c12=1. c13=1. c14=1. c15=1. c16=1. ordered_pair=1. apply=1. relation_dom_restriction=1. cartesian_product2=1. relation_composition=1. set_difference=1. set_intersection2=1. relation_image=1. set_union2=1. relation_inverse_image=1. relation_rng_restriction=1. relation_restriction=1. unordered_pair=1. complements_of_subsets=1. fiber=1. subset_complement=1. meet_of_subsets=1. union_of_subsets=1. f1=1. f2=1. f4=1. f6=1. f7=1. f9=1. f17=1. f22=1. f23=1. f24=1. f25=1. f26=1. f28=1. f35=1. f36=1. f39=1. f47=1. f48=1. f55=1. f56=1. f57=1. f58=1. f59=1. f67=1. f69=1. f71=1. f73=1. f74=1. f75=1. f76=1. f78=1. f79=1. f81=1. f82=1. f84=1. f87=1. f88=1. f89=1. f90=1. f91=1. f92=1. f94=1. f97=1. f99=1. f100=1. relation_dom=1. powerset=1. relation_rng=1. relation_field=1. singleton=1. succ=1. union=1. identity_relation=1. function_inverse=1. relation_inverse=1. set_meet=1. cast_to_subset=1. f3=1. f5=1. f10=1. f14=1. f15=1. f16=1. f18=1. f21=1. f27=1. f29=1. f30=1. f31=1. f40=1. f41=1. f44=1. f45=1. f50=1. f51=1. f52=1. f53=1. f54=1. f65=1. f66=1. f98=1. f105=1. f106=1. unordered_triple=1. subset_difference=1. f8=1. f11=1. f12=1. f19=1. f20=1. f33=1. f34=1. f37=1. f38=1. f43=1. f46=1. f49=1. f62=1. f63=1. f64=1. f68=1. f70=1. f72=1. f77=1. f80=1. f83=1. f85=1. f86=1. f93=1. f95=1. f96=1. f101=1. f102=1. f104=1. f107=1. f108=1. f109=1. f111=1. f112=1. f13=1. f32=1. f42=1. f60=1. f61=1. f103=1. f110=1.
1.00/1.62	
1.00/1.62	============================== end of process initial clauses ========
58.28/58.59	
58.28/58.59	============================== CLAUSES FOR SEARCH ====================
58.28/58.59	
58.28/58.59	============================== end of clauses for search =============
58.28/58.59	
58.28/58.59	============================== SEARCH ================================
58.28/58.59	
58.28/58.59	% Starting search at 0.38 seconds.
58.28/58.59	
58.28/58.59	Low Water (keep): wt=44.000, iters=3487
58.28/58.59	
58.28/58.59	Low Water (keep): wt=40.000, iters=3482
58.28/58.59	
58.28/58.59	Low Water (keep): wt=37.000, iters=3338
58.28/58.59	
58.28/58.59	Low Water (keep): wt=34.000, iters=3337
58.28/58.59	
58.28/58.59	Low Water (keep): wt=32.000, iters=3363
58.28/58.59	
58.28/58.59	Low Water (keep): wt=31.000, iters=3335
58.28/58.59	
58.28/58.59	Low Water (keep): wt=30.000, iters=3352
58.28/58.59	
58.28/58.59	Low Water (keep): wt=29.000, iters=3339
58.28/58.59	
58.28/58.59	Low Water (keep): wt=28.000, iters=3499
58.28/58.59	
58.28/58.59	NOTE: Back_subsumption disabled, ratio of kept to back_subsumed is 217 (0.00 of 1.27 sec).
58.28/58.59	
58.28/58.59	Low Water (keep): wt=27.000, iters=3333
58.28/58.59	
58.28/58.59	Low Water (keep): wt=26.000, iters=3333
58.28/58.59	
58.28/58.59	Low Water (keep): wt=25.000, iters=3366
58.28/58.59	
58.28/58.59	Low Water (keep): wt=24.000, iters=3363
58.28/58.59	
58.28/58.59	Low Water (keep): wt=23.000, iters=3375
58.28/58.59	
58.28/58.59	Low Water (keep): wt=22.000, iters=3340
58.28/58.59	
58.28/58.59	Low Water (keep): wt=21.000, iters=3355
58.28/58.59	
58.28/58.59	Low Water (keep): wt=20.000, iters=3402
58.28/58.59	
58.28/58.59	Low Water (keep): wt=19.000, iters=3384
58.28/58.59	
58.28/58.59	Low Water (keep): wt=17.000, iters=3447
58.28/58.59	
58.28/58.59	Low Water (keep): wt=16.000, iters=3396
58.28/58.59	
58.28/58.59	Low Water (keep): wt=15.000, iters=3373
58.28/58.59	
58.28/58.59	Low Water (keep): wt=14.000, iters=3346
58.28/58.59	
58.28/58.59	Low Water (keep): wt=13.000, iters=3345
58.28/58.59	
58.28/58.59	Low Water (keep): wt=12.000, iters=3386
58.28/58.59	
58.28/58.59	Low Water (keep): wt=11.000, iters=3344
58.28/58.59	
58.28/58.59	Low Water (keep): wt=10.000, iters=3344
58.28/58.59	
58.28/58.59	Low Water (displace): id=3616, wt=72.000
58.28/58.59	
58.28/58.59	Low Water (displace): id=3633, wt=68.000
58.28/58.59	
58.28/58.59	Low Water (displace): id=1925, wt=67.000
58.28/58.59	
58.28/58.59	Low Water (displace): id=4344, wt=66.000
58.28/58.59	
58.28/58.59	Low Water (displace): id=3636, wt=62.000
58.28/58.59	
58.28/58.59	Low Water (displace): id=12458, wt=8.000
58.28/58.59	
58.28/58.59	Low Water (displace): id=12737, wt=7.000
58.28/58.59	
58.28/58.59	Low Water (displace): id=13984, wt=6.000
58.28/58.59	
58.28/58.59	Low Water (keep): wt=9.000, iters=3334
58.28/58.59	
58.28/58.59	Low Water (keep): wt=8.000, iters=3333
58.28/58.59	
58.28/58.59	============================== PROOF =================================
58.28/58.59	% SZS status Theorem
58.28/58.59	% SZS output start Refutation
58.28/58.59	
58.28/58.59	% Proof 1 at 55.92 (+ 1.43) seconds.
58.28/58.59	% Length of proof is 15.
58.28/58.59	% Level of proof is 4.
58.28/58.59	% Maximum clause weight is 12.000.
58.28/58.59	% Given clauses 19046.
58.28/58.59	
58.28/58.59	91 (all A all B all C (relation_of2_as_subset(C,A,B) -> subset(relation_rng(C),B) & subset(relation_dom(C),A))) # label(t12_relset_1) # label(lemma) # label(non_clause).  [assumption].
58.28/58.59	242 (all A all B all C (subset(A,B) & subset(B,C) -> subset(A,C))) # label(t1_xboole_1) # label(lemma) # label(non_clause).  [assumption].
58.28/58.59	268 (all A all B all C all D (relation_of2_as_subset(D,C,A) -> (subset(relation_rng(D),B) -> relation_of2_as_subset(D,C,B)))) # label(t14_relset_1) # label(lemma) # label(non_clause).  [assumption].
58.28/58.59	325 -(all A all B all C all D (relation_of2_as_subset(D,C,A) -> (subset(A,B) -> relation_of2_as_subset(D,C,B)))) # label(t16_relset_1) # label(negated_conjecture) # label(non_clause).  [assumption].
58.28/58.59	550 -relation_of2_as_subset(A,B,C) | subset(relation_rng(A),C) # label(t12_relset_1) # label(lemma).  [clausify(91)].
58.28/58.59	879 -subset(A,B) | -subset(B,C) | subset(A,C) # label(t1_xboole_1) # label(lemma).  [clausify(242)].
58.28/58.59	951 -relation_of2_as_subset(A,B,C) | -subset(relation_rng(A),D) | relation_of2_as_subset(A,B,D) # label(t14_relset_1) # label(lemma).  [clausify(268)].
58.28/58.59	1081 relation_of2_as_subset(c16,c15,c13) # label(t16_relset_1) # label(negated_conjecture).  [clausify(325)].
58.28/58.59	1082 subset(c13,c14) # label(t16_relset_1) # label(negated_conjecture).  [clausify(325)].
58.28/58.59	1083 -relation_of2_as_subset(c16,c15,c14) # label(t16_relset_1) # label(negated_conjecture).  [clausify(325)].
58.28/58.59	5650 -subset(relation_rng(c16),A) | relation_of2_as_subset(c16,c15,A).  [resolve(1081,a,951,a)].
58.28/58.59	5652 subset(relation_rng(c16),c13).  [resolve(1081,a,550,a)].
58.28/58.59	5663 -subset(A,c13) | subset(A,c14).  [resolve(1082,a,879,b)].
58.28/58.59	35053 subset(relation_rng(c16),c14).  [resolve(5663,a,5652,a)].
58.28/58.59	160648 $F.  [resolve(5650,a,35053,a),unit_del(a,1083)].
58.28/58.59	
58.28/58.59	% SZS output end Refutation
58.28/58.59	============================== end of proof ==========================
58.28/58.59	
58.28/58.59	============================== STATISTICS ============================
58.28/58.59	
58.28/58.59	Given=19046. Generated=2527289. Kept=160162. proofs=1.
58.28/58.59	Usable=18514. Sos=9886. Demods=914. Limbo=8, Disabled=132481. Hints=0.
58.28/58.59	Megabytes=124.30.
58.28/58.59	User_CPU=55.92, System_CPU=1.43, Wall_clock=58.
58.28/58.59	
58.28/58.59	============================== end of statistics =====================
58.28/58.59	
58.28/58.59	============================== end of search =========================
58.28/58.59	
58.28/58.59	THEOREM PROVED
58.28/58.59	% SZS status Theorem
58.28/58.59	
58.28/58.59	Exiting with 1 proof.
58.28/58.59	
58.28/58.59	Process 7407 exit (max_proofs) Thu Jul  2 07:44:57 2020
58.28/58.59	Prover9 interrupted
58.28/58.59	EOF