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Theorem rankcf 10815
Description: Any set must be at least as large as the cofinality of its rank, because the ranks of the elements of 𝐴 form a cofinal map into (rank‘𝐴). (Contributed by Mario Carneiro, 27-May-2013.)
Assertion
Ref Expression
rankcf ¬ 𝐴 ≺ (cf‘(rank‘𝐴))

Proof of Theorem rankcf
Dummy variables 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rankon 9833 . . 3 (rank‘𝐴) ∈ On
2 onzsl 7867 . . 3 ((rank‘𝐴) ∈ On ↔ ((rank‘𝐴) = ∅ ∨ ∃𝑥 ∈ On (rank‘𝐴) = suc 𝑥 ∨ ((rank‘𝐴) ∈ V ∧ Lim (rank‘𝐴))))
31, 2mpbi 230 . 2 ((rank‘𝐴) = ∅ ∨ ∃𝑥 ∈ On (rank‘𝐴) = suc 𝑥 ∨ ((rank‘𝐴) ∈ V ∧ Lim (rank‘𝐴)))
4 sdom0 9147 . . . 4 ¬ 𝐴 ≺ ∅
5 fveq2 6907 . . . . . 6 ((rank‘𝐴) = ∅ → (cf‘(rank‘𝐴)) = (cf‘∅))
6 cf0 10289 . . . . . 6 (cf‘∅) = ∅
75, 6eqtrdi 2791 . . . . 5 ((rank‘𝐴) = ∅ → (cf‘(rank‘𝐴)) = ∅)
87breq2d 5160 . . . 4 ((rank‘𝐴) = ∅ → (𝐴 ≺ (cf‘(rank‘𝐴)) ↔ 𝐴 ≺ ∅))
94, 8mtbiri 327 . . 3 ((rank‘𝐴) = ∅ → ¬ 𝐴 ≺ (cf‘(rank‘𝐴)))
10 fveq2 6907 . . . . . . 7 ((rank‘𝐴) = suc 𝑥 → (cf‘(rank‘𝐴)) = (cf‘suc 𝑥))
11 cfsuc 10295 . . . . . . 7 (𝑥 ∈ On → (cf‘suc 𝑥) = 1o)
1210, 11sylan9eqr 2797 . . . . . 6 ((𝑥 ∈ On ∧ (rank‘𝐴) = suc 𝑥) → (cf‘(rank‘𝐴)) = 1o)
13 nsuceq0 6469 . . . . . . . . 9 suc 𝑥 ≠ ∅
14 neeq1 3001 . . . . . . . . 9 ((rank‘𝐴) = suc 𝑥 → ((rank‘𝐴) ≠ ∅ ↔ suc 𝑥 ≠ ∅))
1513, 14mpbiri 258 . . . . . . . 8 ((rank‘𝐴) = suc 𝑥 → (rank‘𝐴) ≠ ∅)
16 fveq2 6907 . . . . . . . . . . 11 (𝐴 = ∅ → (rank‘𝐴) = (rank‘∅))
17 0elon 6440 . . . . . . . . . . . . 13 ∅ ∈ On
18 r1fnon 9805 . . . . . . . . . . . . . 14 𝑅1 Fn On
1918fndmi 6673 . . . . . . . . . . . . 13 dom 𝑅1 = On
2017, 19eleqtrri 2838 . . . . . . . . . . . 12 ∅ ∈ dom 𝑅1
21 rankonid 9867 . . . . . . . . . . . 12 (∅ ∈ dom 𝑅1 ↔ (rank‘∅) = ∅)
2220, 21mpbi 230 . . . . . . . . . . 11 (rank‘∅) = ∅
2316, 22eqtrdi 2791 . . . . . . . . . 10 (𝐴 = ∅ → (rank‘𝐴) = ∅)
2423necon3i 2971 . . . . . . . . 9 ((rank‘𝐴) ≠ ∅ → 𝐴 ≠ ∅)
25 rankvaln 9837 . . . . . . . . . . 11 𝐴 (𝑅1 “ On) → (rank‘𝐴) = ∅)
2625necon1ai 2966 . . . . . . . . . 10 ((rank‘𝐴) ≠ ∅ → 𝐴 (𝑅1 “ On))
27 breq2 5152 . . . . . . . . . . 11 (𝑦 = 𝐴 → (1o𝑦 ↔ 1o𝐴))
28 neeq1 3001 . . . . . . . . . . 11 (𝑦 = 𝐴 → (𝑦 ≠ ∅ ↔ 𝐴 ≠ ∅))
29 0sdom1dom 9272 . . . . . . . . . . . 12 (∅ ≺ 𝑦 ↔ 1o𝑦)
30 vex 3482 . . . . . . . . . . . . 13 𝑦 ∈ V
31300sdom 9146 . . . . . . . . . . . 12 (∅ ≺ 𝑦𝑦 ≠ ∅)
3229, 31bitr3i 277 . . . . . . . . . . 11 (1o𝑦𝑦 ≠ ∅)
3327, 28, 32vtoclbg 3557 . . . . . . . . . 10 (𝐴 (𝑅1 “ On) → (1o𝐴𝐴 ≠ ∅))
3426, 33syl 17 . . . . . . . . 9 ((rank‘𝐴) ≠ ∅ → (1o𝐴𝐴 ≠ ∅))
3524, 34mpbird 257 . . . . . . . 8 ((rank‘𝐴) ≠ ∅ → 1o𝐴)
3615, 35syl 17 . . . . . . 7 ((rank‘𝐴) = suc 𝑥 → 1o𝐴)
3736adantl 481 . . . . . 6 ((𝑥 ∈ On ∧ (rank‘𝐴) = suc 𝑥) → 1o𝐴)
3812, 37eqbrtrd 5170 . . . . 5 ((𝑥 ∈ On ∧ (rank‘𝐴) = suc 𝑥) → (cf‘(rank‘𝐴)) ≼ 𝐴)
3938rexlimiva 3145 . . . 4 (∃𝑥 ∈ On (rank‘𝐴) = suc 𝑥 → (cf‘(rank‘𝐴)) ≼ 𝐴)
40 domnsym 9138 . . . 4 ((cf‘(rank‘𝐴)) ≼ 𝐴 → ¬ 𝐴 ≺ (cf‘(rank‘𝐴)))
4139, 40syl 17 . . 3 (∃𝑥 ∈ On (rank‘𝐴) = suc 𝑥 → ¬ 𝐴 ≺ (cf‘(rank‘𝐴)))
42 nlim0 6445 . . . . . . . . . . . . . . . . 17 ¬ Lim ∅
43 limeq 6398 . . . . . . . . . . . . . . . . 17 ((rank‘𝐴) = ∅ → (Lim (rank‘𝐴) ↔ Lim ∅))
4442, 43mtbiri 327 . . . . . . . . . . . . . . . 16 ((rank‘𝐴) = ∅ → ¬ Lim (rank‘𝐴))
4525, 44syl 17 . . . . . . . . . . . . . . 15 𝐴 (𝑅1 “ On) → ¬ Lim (rank‘𝐴))
4645con4i 114 . . . . . . . . . . . . . 14 (Lim (rank‘𝐴) → 𝐴 (𝑅1 “ On))
47 r1elssi 9843 . . . . . . . . . . . . . 14 (𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
4846, 47syl 17 . . . . . . . . . . . . 13 (Lim (rank‘𝐴) → 𝐴 (𝑅1 “ On))
4948sselda 3995 . . . . . . . . . . . 12 ((Lim (rank‘𝐴) ∧ 𝑥𝐴) → 𝑥 (𝑅1 “ On))
50 ranksnb 9865 . . . . . . . . . . . 12 (𝑥 (𝑅1 “ On) → (rank‘{𝑥}) = suc (rank‘𝑥))
5149, 50syl 17 . . . . . . . . . . 11 ((Lim (rank‘𝐴) ∧ 𝑥𝐴) → (rank‘{𝑥}) = suc (rank‘𝑥))
52 rankelb 9862 . . . . . . . . . . . . . 14 (𝐴 (𝑅1 “ On) → (𝑥𝐴 → (rank‘𝑥) ∈ (rank‘𝐴)))
5346, 52syl 17 . . . . . . . . . . . . 13 (Lim (rank‘𝐴) → (𝑥𝐴 → (rank‘𝑥) ∈ (rank‘𝐴)))
54 limsuc 7870 . . . . . . . . . . . . 13 (Lim (rank‘𝐴) → ((rank‘𝑥) ∈ (rank‘𝐴) ↔ suc (rank‘𝑥) ∈ (rank‘𝐴)))
5553, 54sylibd 239 . . . . . . . . . . . 12 (Lim (rank‘𝐴) → (𝑥𝐴 → suc (rank‘𝑥) ∈ (rank‘𝐴)))
5655imp 406 . . . . . . . . . . 11 ((Lim (rank‘𝐴) ∧ 𝑥𝐴) → suc (rank‘𝑥) ∈ (rank‘𝐴))
5751, 56eqeltrd 2839 . . . . . . . . . 10 ((Lim (rank‘𝐴) ∧ 𝑥𝐴) → (rank‘{𝑥}) ∈ (rank‘𝐴))
58 eleq1a 2834 . . . . . . . . . 10 ((rank‘{𝑥}) ∈ (rank‘𝐴) → (𝑤 = (rank‘{𝑥}) → 𝑤 ∈ (rank‘𝐴)))
5957, 58syl 17 . . . . . . . . 9 ((Lim (rank‘𝐴) ∧ 𝑥𝐴) → (𝑤 = (rank‘{𝑥}) → 𝑤 ∈ (rank‘𝐴)))
6059rexlimdva 3153 . . . . . . . 8 (Lim (rank‘𝐴) → (∃𝑥𝐴 𝑤 = (rank‘{𝑥}) → 𝑤 ∈ (rank‘𝐴)))
6160abssdv 4078 . . . . . . 7 (Lim (rank‘𝐴) → {𝑤 ∣ ∃𝑥𝐴 𝑤 = (rank‘{𝑥})} ⊆ (rank‘𝐴))
62 vsnex 5440 . . . . . . . . . . . . 13 {𝑥} ∈ V
6362dfiun2 5038 . . . . . . . . . . . 12 𝑥𝐴 {𝑥} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}}
64 iunid 5065 . . . . . . . . . . . 12 𝑥𝐴 {𝑥} = 𝐴
6563, 64eqtr3i 2765 . . . . . . . . . . 11 {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}} = 𝐴
6665fveq2i 6910 . . . . . . . . . 10 (rank‘ {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}}) = (rank‘𝐴)
6747sselda 3995 . . . . . . . . . . . . . . 15 ((𝐴 (𝑅1 “ On) ∧ 𝑥𝐴) → 𝑥 (𝑅1 “ On))
68 snwf 9847 . . . . . . . . . . . . . . 15 (𝑥 (𝑅1 “ On) → {𝑥} ∈ (𝑅1 “ On))
69 eleq1a 2834 . . . . . . . . . . . . . . 15 ({𝑥} ∈ (𝑅1 “ On) → (𝑦 = {𝑥} → 𝑦 (𝑅1 “ On)))
7067, 68, 693syl 18 . . . . . . . . . . . . . 14 ((𝐴 (𝑅1 “ On) ∧ 𝑥𝐴) → (𝑦 = {𝑥} → 𝑦 (𝑅1 “ On)))
7170rexlimdva 3153 . . . . . . . . . . . . 13 (𝐴 (𝑅1 “ On) → (∃𝑥𝐴 𝑦 = {𝑥} → 𝑦 (𝑅1 “ On)))
7271abssdv 4078 . . . . . . . . . . . 12 (𝐴 (𝑅1 “ On) → {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}} ⊆ (𝑅1 “ On))
73 abrexexg 7984 . . . . . . . . . . . . 13 (𝐴 (𝑅1 “ On) → {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}} ∈ V)
74 eleq1 2827 . . . . . . . . . . . . . 14 (𝑧 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}} → (𝑧 (𝑅1 “ On) ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}} ∈ (𝑅1 “ On)))
75 sseq1 4021 . . . . . . . . . . . . . 14 (𝑧 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}} → (𝑧 (𝑅1 “ On) ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}} ⊆ (𝑅1 “ On)))
76 vex 3482 . . . . . . . . . . . . . . 15 𝑧 ∈ V
7776r1elss 9844 . . . . . . . . . . . . . 14 (𝑧 (𝑅1 “ On) ↔ 𝑧 (𝑅1 “ On))
7874, 75, 77vtoclbg 3557 . . . . . . . . . . . . 13 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}} ∈ V → ({𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}} ∈ (𝑅1 “ On) ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}} ⊆ (𝑅1 “ On)))
7973, 78syl 17 . . . . . . . . . . . 12 (𝐴 (𝑅1 “ On) → ({𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}} ∈ (𝑅1 “ On) ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}} ⊆ (𝑅1 “ On)))
8072, 79mpbird 257 . . . . . . . . . . 11 (𝐴 (𝑅1 “ On) → {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}} ∈ (𝑅1 “ On))
81 rankuni2b 9891 . . . . . . . . . . 11 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}} ∈ (𝑅1 “ On) → (rank‘ {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}}) = 𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}} (rank‘𝑧))
8280, 81syl 17 . . . . . . . . . 10 (𝐴 (𝑅1 “ On) → (rank‘ {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}}) = 𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}} (rank‘𝑧))
8366, 82eqtr3id 2789 . . . . . . . . 9 (𝐴 (𝑅1 “ On) → (rank‘𝐴) = 𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}} (rank‘𝑧))
84 fvex 6920 . . . . . . . . . . 11 (rank‘𝑧) ∈ V
8584dfiun2 5038 . . . . . . . . . 10 𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}} (rank‘𝑧) = {𝑤 ∣ ∃𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}}𝑤 = (rank‘𝑧)}
86 fveq2 6907 . . . . . . . . . . . 12 (𝑧 = {𝑥} → (rank‘𝑧) = (rank‘{𝑥}))
8762, 86abrexco 7264 . . . . . . . . . . 11 {𝑤 ∣ ∃𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}}𝑤 = (rank‘𝑧)} = {𝑤 ∣ ∃𝑥𝐴 𝑤 = (rank‘{𝑥})}
8887unieqi 4924 . . . . . . . . . 10 {𝑤 ∣ ∃𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}}𝑤 = (rank‘𝑧)} = {𝑤 ∣ ∃𝑥𝐴 𝑤 = (rank‘{𝑥})}
8985, 88eqtri 2763 . . . . . . . . 9 𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}} (rank‘𝑧) = {𝑤 ∣ ∃𝑥𝐴 𝑤 = (rank‘{𝑥})}
9083, 89eqtr2di 2792 . . . . . . . 8 (𝐴 (𝑅1 “ On) → {𝑤 ∣ ∃𝑥𝐴 𝑤 = (rank‘{𝑥})} = (rank‘𝐴))
9146, 90syl 17 . . . . . . 7 (Lim (rank‘𝐴) → {𝑤 ∣ ∃𝑥𝐴 𝑤 = (rank‘{𝑥})} = (rank‘𝐴))
92 fvex 6920 . . . . . . . 8 (rank‘𝐴) ∈ V
9392cfslb 10304 . . . . . . 7 ((Lim (rank‘𝐴) ∧ {𝑤 ∣ ∃𝑥𝐴 𝑤 = (rank‘{𝑥})} ⊆ (rank‘𝐴) ∧ {𝑤 ∣ ∃𝑥𝐴 𝑤 = (rank‘{𝑥})} = (rank‘𝐴)) → (cf‘(rank‘𝐴)) ≼ {𝑤 ∣ ∃𝑥𝐴 𝑤 = (rank‘{𝑥})})
9461, 91, 93mpd3an23 1462 . . . . . 6 (Lim (rank‘𝐴) → (cf‘(rank‘𝐴)) ≼ {𝑤 ∣ ∃𝑥𝐴 𝑤 = (rank‘{𝑥})})
95 2fveq3 6912 . . . . . . . . . 10 (𝑦 = 𝐴 → (cf‘(rank‘𝑦)) = (cf‘(rank‘𝐴)))
96 breq12 5153 . . . . . . . . . 10 ((𝑦 = 𝐴 ∧ (cf‘(rank‘𝑦)) = (cf‘(rank‘𝐴))) → (𝑦 ≺ (cf‘(rank‘𝑦)) ↔ 𝐴 ≺ (cf‘(rank‘𝐴))))
9795, 96mpdan 687 . . . . . . . . 9 (𝑦 = 𝐴 → (𝑦 ≺ (cf‘(rank‘𝑦)) ↔ 𝐴 ≺ (cf‘(rank‘𝐴))))
98 rexeq 3320 . . . . . . . . . . 11 (𝑦 = 𝐴 → (∃𝑥𝑦 𝑤 = (rank‘{𝑥}) ↔ ∃𝑥𝐴 𝑤 = (rank‘{𝑥})))
9998abbidv 2806 . . . . . . . . . 10 (𝑦 = 𝐴 → {𝑤 ∣ ∃𝑥𝑦 𝑤 = (rank‘{𝑥})} = {𝑤 ∣ ∃𝑥𝐴 𝑤 = (rank‘{𝑥})})
100 breq12 5153 . . . . . . . . . 10 (({𝑤 ∣ ∃𝑥𝑦 𝑤 = (rank‘{𝑥})} = {𝑤 ∣ ∃𝑥𝐴 𝑤 = (rank‘{𝑥})} ∧ 𝑦 = 𝐴) → ({𝑤 ∣ ∃𝑥𝑦 𝑤 = (rank‘{𝑥})} ≼ 𝑦 ↔ {𝑤 ∣ ∃𝑥𝐴 𝑤 = (rank‘{𝑥})} ≼ 𝐴))
10199, 100mpancom 688 . . . . . . . . 9 (𝑦 = 𝐴 → ({𝑤 ∣ ∃𝑥𝑦 𝑤 = (rank‘{𝑥})} ≼ 𝑦 ↔ {𝑤 ∣ ∃𝑥𝐴 𝑤 = (rank‘{𝑥})} ≼ 𝐴))
10297, 101imbi12d 344 . . . . . . . 8 (𝑦 = 𝐴 → ((𝑦 ≺ (cf‘(rank‘𝑦)) → {𝑤 ∣ ∃𝑥𝑦 𝑤 = (rank‘{𝑥})} ≼ 𝑦) ↔ (𝐴 ≺ (cf‘(rank‘𝐴)) → {𝑤 ∣ ∃𝑥𝐴 𝑤 = (rank‘{𝑥})} ≼ 𝐴)))
103 eqid 2735 . . . . . . . . . 10 (𝑥𝑦 ↦ (rank‘{𝑥})) = (𝑥𝑦 ↦ (rank‘{𝑥}))
104103rnmpt 5971 . . . . . . . . 9 ran (𝑥𝑦 ↦ (rank‘{𝑥})) = {𝑤 ∣ ∃𝑥𝑦 𝑤 = (rank‘{𝑥})}
105 cfon 10293 . . . . . . . . . . 11 (cf‘(rank‘𝑦)) ∈ On
106 sdomdom 9019 . . . . . . . . . . 11 (𝑦 ≺ (cf‘(rank‘𝑦)) → 𝑦 ≼ (cf‘(rank‘𝑦)))
107 ondomen 10075 . . . . . . . . . . 11 (((cf‘(rank‘𝑦)) ∈ On ∧ 𝑦 ≼ (cf‘(rank‘𝑦))) → 𝑦 ∈ dom card)
108105, 106, 107sylancr 587 . . . . . . . . . 10 (𝑦 ≺ (cf‘(rank‘𝑦)) → 𝑦 ∈ dom card)
109 fvex 6920 . . . . . . . . . . . 12 (rank‘{𝑥}) ∈ V
110109, 103fnmpti 6712 . . . . . . . . . . 11 (𝑥𝑦 ↦ (rank‘{𝑥})) Fn 𝑦
111 dffn4 6827 . . . . . . . . . . 11 ((𝑥𝑦 ↦ (rank‘{𝑥})) Fn 𝑦 ↔ (𝑥𝑦 ↦ (rank‘{𝑥})):𝑦onto→ran (𝑥𝑦 ↦ (rank‘{𝑥})))
112110, 111mpbi 230 . . . . . . . . . 10 (𝑥𝑦 ↦ (rank‘{𝑥})):𝑦onto→ran (𝑥𝑦 ↦ (rank‘{𝑥}))
113 fodomnum 10095 . . . . . . . . . 10 (𝑦 ∈ dom card → ((𝑥𝑦 ↦ (rank‘{𝑥})):𝑦onto→ran (𝑥𝑦 ↦ (rank‘{𝑥})) → ran (𝑥𝑦 ↦ (rank‘{𝑥})) ≼ 𝑦))
114108, 112, 113mpisyl 21 . . . . . . . . 9 (𝑦 ≺ (cf‘(rank‘𝑦)) → ran (𝑥𝑦 ↦ (rank‘{𝑥})) ≼ 𝑦)
115104, 114eqbrtrrid 5184 . . . . . . . 8 (𝑦 ≺ (cf‘(rank‘𝑦)) → {𝑤 ∣ ∃𝑥𝑦 𝑤 = (rank‘{𝑥})} ≼ 𝑦)
116102, 115vtoclg 3554 . . . . . . 7 (𝐴 (𝑅1 “ On) → (𝐴 ≺ (cf‘(rank‘𝐴)) → {𝑤 ∣ ∃𝑥𝐴 𝑤 = (rank‘{𝑥})} ≼ 𝐴))
11746, 116syl 17 . . . . . 6 (Lim (rank‘𝐴) → (𝐴 ≺ (cf‘(rank‘𝐴)) → {𝑤 ∣ ∃𝑥𝐴 𝑤 = (rank‘{𝑥})} ≼ 𝐴))
118 domtr 9046 . . . . . . 7 (((cf‘(rank‘𝐴)) ≼ {𝑤 ∣ ∃𝑥𝐴 𝑤 = (rank‘{𝑥})} ∧ {𝑤 ∣ ∃𝑥𝐴 𝑤 = (rank‘{𝑥})} ≼ 𝐴) → (cf‘(rank‘𝐴)) ≼ 𝐴)
119118, 40syl 17 . . . . . 6 (((cf‘(rank‘𝐴)) ≼ {𝑤 ∣ ∃𝑥𝐴 𝑤 = (rank‘{𝑥})} ∧ {𝑤 ∣ ∃𝑥𝐴 𝑤 = (rank‘{𝑥})} ≼ 𝐴) → ¬ 𝐴 ≺ (cf‘(rank‘𝐴)))
12094, 117, 119syl6an 684 . . . . 5 (Lim (rank‘𝐴) → (𝐴 ≺ (cf‘(rank‘𝐴)) → ¬ 𝐴 ≺ (cf‘(rank‘𝐴))))
121120pm2.01d 190 . . . 4 (Lim (rank‘𝐴) → ¬ 𝐴 ≺ (cf‘(rank‘𝐴)))
122121adantl 481 . . 3 (((rank‘𝐴) ∈ V ∧ Lim (rank‘𝐴)) → ¬ 𝐴 ≺ (cf‘(rank‘𝐴)))
1239, 41, 1223jaoi 1427 . 2 (((rank‘𝐴) = ∅ ∨ ∃𝑥 ∈ On (rank‘𝐴) = suc 𝑥 ∨ ((rank‘𝐴) ∈ V ∧ Lim (rank‘𝐴))) → ¬ 𝐴 ≺ (cf‘(rank‘𝐴)))
1243, 123ax-mp 5 1 ¬ 𝐴 ≺ (cf‘(rank‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3o 1085   = wceq 1537  wcel 2106  {cab 2712  wne 2938  wrex 3068  Vcvv 3478  wss 3963  c0 4339  {csn 4631   cuni 4912   ciun 4996   class class class wbr 5148  cmpt 5231  dom cdm 5689  ran crn 5690  cima 5692  Oncon0 6386  Lim wlim 6387  suc csuc 6388   Fn wfn 6558  ontowfo 6561  cfv 6563  1oc1o 8498  cdom 8982  csdm 8983  𝑅1cr1 9800  rankcrnk 9801  cardccrd 9973  cfccf 9975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-r1 9802  df-rank 9803  df-card 9977  df-cf 9979  df-acn 9980
This theorem is referenced by:  inatsk  10816  grur1  10858
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