MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rankcf Structured version   Visualization version   GIF version

Theorem rankcf 10768
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 9786 . . 3 (rank‘𝐴) ∈ On
2 onzsl 7828 . . 3 ((rank‘𝐴) ∈ On ↔ ((rank‘𝐴) = ∅ ∨ ∃𝑥 ∈ On (rank‘𝐴) = suc 𝑥 ∨ ((rank‘𝐴) ∈ V ∧ Lim (rank‘𝐴))))
31, 2mpbi 229 . 2 ((rank‘𝐴) = ∅ ∨ ∃𝑥 ∈ On (rank‘𝐴) = suc 𝑥 ∨ ((rank‘𝐴) ∈ V ∧ Lim (rank‘𝐴)))
4 sdom0 9104 . . . 4 ¬ 𝐴 ≺ ∅
5 fveq2 6881 . . . . . 6 ((rank‘𝐴) = ∅ → (cf‘(rank‘𝐴)) = (cf‘∅))
6 cf0 10242 . . . . . 6 (cf‘∅) = ∅
75, 6eqtrdi 2780 . . . . 5 ((rank‘𝐴) = ∅ → (cf‘(rank‘𝐴)) = ∅)
87breq2d 5150 . . . 4 ((rank‘𝐴) = ∅ → (𝐴 ≺ (cf‘(rank‘𝐴)) ↔ 𝐴 ≺ ∅))
94, 8mtbiri 327 . . 3 ((rank‘𝐴) = ∅ → ¬ 𝐴 ≺ (cf‘(rank‘𝐴)))
10 fveq2 6881 . . . . . . 7 ((rank‘𝐴) = suc 𝑥 → (cf‘(rank‘𝐴)) = (cf‘suc 𝑥))
11 cfsuc 10248 . . . . . . 7 (𝑥 ∈ On → (cf‘suc 𝑥) = 1o)
1210, 11sylan9eqr 2786 . . . . . 6 ((𝑥 ∈ On ∧ (rank‘𝐴) = suc 𝑥) → (cf‘(rank‘𝐴)) = 1o)
13 nsuceq0 6437 . . . . . . . . 9 suc 𝑥 ≠ ∅
14 neeq1 2995 . . . . . . . . 9 ((rank‘𝐴) = suc 𝑥 → ((rank‘𝐴) ≠ ∅ ↔ suc 𝑥 ≠ ∅))
1513, 14mpbiri 258 . . . . . . . 8 ((rank‘𝐴) = suc 𝑥 → (rank‘𝐴) ≠ ∅)
16 fveq2 6881 . . . . . . . . . . 11 (𝐴 = ∅ → (rank‘𝐴) = (rank‘∅))
17 0elon 6408 . . . . . . . . . . . . 13 ∅ ∈ On
18 r1fnon 9758 . . . . . . . . . . . . . 14 𝑅1 Fn On
1918fndmi 6643 . . . . . . . . . . . . 13 dom 𝑅1 = On
2017, 19eleqtrri 2824 . . . . . . . . . . . 12 ∅ ∈ dom 𝑅1
21 rankonid 9820 . . . . . . . . . . . 12 (∅ ∈ dom 𝑅1 ↔ (rank‘∅) = ∅)
2220, 21mpbi 229 . . . . . . . . . . 11 (rank‘∅) = ∅
2316, 22eqtrdi 2780 . . . . . . . . . 10 (𝐴 = ∅ → (rank‘𝐴) = ∅)
2423necon3i 2965 . . . . . . . . 9 ((rank‘𝐴) ≠ ∅ → 𝐴 ≠ ∅)
25 rankvaln 9790 . . . . . . . . . . 11 𝐴 (𝑅1 “ On) → (rank‘𝐴) = ∅)
2625necon1ai 2960 . . . . . . . . . 10 ((rank‘𝐴) ≠ ∅ → 𝐴 (𝑅1 “ On))
27 breq2 5142 . . . . . . . . . . 11 (𝑦 = 𝐴 → (1o𝑦 ↔ 1o𝐴))
28 neeq1 2995 . . . . . . . . . . 11 (𝑦 = 𝐴 → (𝑦 ≠ ∅ ↔ 𝐴 ≠ ∅))
29 0sdom1dom 9234 . . . . . . . . . . . 12 (∅ ≺ 𝑦 ↔ 1o𝑦)
30 vex 3470 . . . . . . . . . . . . 13 𝑦 ∈ V
31300sdom 9103 . . . . . . . . . . . 12 (∅ ≺ 𝑦𝑦 ≠ ∅)
3229, 31bitr3i 277 . . . . . . . . . . 11 (1o𝑦𝑦 ≠ ∅)
3327, 28, 32vtoclbg 3537 . . . . . . . . . 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 5160 . . . . 5 ((𝑥 ∈ On ∧ (rank‘𝐴) = suc 𝑥) → (cf‘(rank‘𝐴)) ≼ 𝐴)
3938rexlimiva 3139 . . . 4 (∃𝑥 ∈ On (rank‘𝐴) = suc 𝑥 → (cf‘(rank‘𝐴)) ≼ 𝐴)
40 domnsym 9095 . . . 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 6413 . . . . . . . . . . . . . . . . 17 ¬ Lim ∅
43 limeq 6366 . . . . . . . . . . . . . . . . 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 9796 . . . . . . . . . . . . . 14 (𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
4846, 47syl 17 . . . . . . . . . . . . 13 (Lim (rank‘𝐴) → 𝐴 (𝑅1 “ On))
4948sselda 3974 . . . . . . . . . . . 12 ((Lim (rank‘𝐴) ∧ 𝑥𝐴) → 𝑥 (𝑅1 “ On))
50 ranksnb 9818 . . . . . . . . . . . 12 (𝑥 (𝑅1 “ On) → (rank‘{𝑥}) = suc (rank‘𝑥))
5149, 50syl 17 . . . . . . . . . . 11 ((Lim (rank‘𝐴) ∧ 𝑥𝐴) → (rank‘{𝑥}) = suc (rank‘𝑥))
52 rankelb 9815 . . . . . . . . . . . . . 14 (𝐴 (𝑅1 “ On) → (𝑥𝐴 → (rank‘𝑥) ∈ (rank‘𝐴)))
5346, 52syl 17 . . . . . . . . . . . . 13 (Lim (rank‘𝐴) → (𝑥𝐴 → (rank‘𝑥) ∈ (rank‘𝐴)))
54 limsuc 7831 . . . . . . . . . . . . 13 (Lim (rank‘𝐴) → ((rank‘𝑥) ∈ (rank‘𝐴) ↔ suc (rank‘𝑥) ∈ (rank‘𝐴)))
5553, 54sylibd 238 . . . . . . . . . . . 12 (Lim (rank‘𝐴) → (𝑥𝐴 → suc (rank‘𝑥) ∈ (rank‘𝐴)))
5655imp 406 . . . . . . . . . . 11 ((Lim (rank‘𝐴) ∧ 𝑥𝐴) → suc (rank‘𝑥) ∈ (rank‘𝐴))
5751, 56eqeltrd 2825 . . . . . . . . . 10 ((Lim (rank‘𝐴) ∧ 𝑥𝐴) → (rank‘{𝑥}) ∈ (rank‘𝐴))
58 eleq1a 2820 . . . . . . . . . 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 3147 . . . . . . . 8 (Lim (rank‘𝐴) → (∃𝑥𝐴 𝑤 = (rank‘{𝑥}) → 𝑤 ∈ (rank‘𝐴)))
6160abssdv 4057 . . . . . . 7 (Lim (rank‘𝐴) → {𝑤 ∣ ∃𝑥𝐴 𝑤 = (rank‘{𝑥})} ⊆ (rank‘𝐴))
62 vsnex 5419 . . . . . . . . . . . . 13 {𝑥} ∈ V
6362dfiun2 5026 . . . . . . . . . . . 12 𝑥𝐴 {𝑥} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}}
64 iunid 5053 . . . . . . . . . . . 12 𝑥𝐴 {𝑥} = 𝐴
6563, 64eqtr3i 2754 . . . . . . . . . . 11 {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}} = 𝐴
6665fveq2i 6884 . . . . . . . . . 10 (rank‘ {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}}) = (rank‘𝐴)
6747sselda 3974 . . . . . . . . . . . . . . 15 ((𝐴 (𝑅1 “ On) ∧ 𝑥𝐴) → 𝑥 (𝑅1 “ On))
68 snwf 9800 . . . . . . . . . . . . . . 15 (𝑥 (𝑅1 “ On) → {𝑥} ∈ (𝑅1 “ On))
69 eleq1a 2820 . . . . . . . . . . . . . . 15 ({𝑥} ∈ (𝑅1 “ On) → (𝑦 = {𝑥} → 𝑦 (𝑅1 “ On)))
7067, 68, 693syl 18 . . . . . . . . . . . . . 14 ((𝐴 (𝑅1 “ On) ∧ 𝑥𝐴) → (𝑦 = {𝑥} → 𝑦 (𝑅1 “ On)))
7170rexlimdva 3147 . . . . . . . . . . . . 13 (𝐴 (𝑅1 “ On) → (∃𝑥𝐴 𝑦 = {𝑥} → 𝑦 (𝑅1 “ On)))
7271abssdv 4057 . . . . . . . . . . . 12 (𝐴 (𝑅1 “ On) → {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}} ⊆ (𝑅1 “ On))
73 abrexexg 7940 . . . . . . . . . . . . 13 (𝐴 (𝑅1 “ On) → {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}} ∈ V)
74 eleq1 2813 . . . . . . . . . . . . . 14 (𝑧 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}} → (𝑧 (𝑅1 “ On) ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}} ∈ (𝑅1 “ On)))
75 sseq1 3999 . . . . . . . . . . . . . 14 (𝑧 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}} → (𝑧 (𝑅1 “ On) ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}} ⊆ (𝑅1 “ On)))
76 vex 3470 . . . . . . . . . . . . . . 15 𝑧 ∈ V
7776r1elss 9797 . . . . . . . . . . . . . 14 (𝑧 (𝑅1 “ On) ↔ 𝑧 (𝑅1 “ On))
7874, 75, 77vtoclbg 3537 . . . . . . . . . . . . 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 9844 . . . . . . . . . . 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 2778 . . . . . . . . 9 (𝐴 (𝑅1 “ On) → (rank‘𝐴) = 𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}} (rank‘𝑧))
84 fvex 6894 . . . . . . . . . . 11 (rank‘𝑧) ∈ V
8584dfiun2 5026 . . . . . . . . . 10 𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}} (rank‘𝑧) = {𝑤 ∣ ∃𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}}𝑤 = (rank‘𝑧)}
86 fveq2 6881 . . . . . . . . . . . 12 (𝑧 = {𝑥} → (rank‘𝑧) = (rank‘{𝑥}))
8762, 86abrexco 7235 . . . . . . . . . . 11 {𝑤 ∣ ∃𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}}𝑤 = (rank‘𝑧)} = {𝑤 ∣ ∃𝑥𝐴 𝑤 = (rank‘{𝑥})}
8887unieqi 4911 . . . . . . . . . 10 {𝑤 ∣ ∃𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}}𝑤 = (rank‘𝑧)} = {𝑤 ∣ ∃𝑥𝐴 𝑤 = (rank‘{𝑥})}
8985, 88eqtri 2752 . . . . . . . . 9 𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = {𝑥}} (rank‘𝑧) = {𝑤 ∣ ∃𝑥𝐴 𝑤 = (rank‘{𝑥})}
9083, 89eqtr2di 2781 . . . . . . . 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 6894 . . . . . . . 8 (rank‘𝐴) ∈ V
9392cfslb 10257 . . . . . . 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 1459 . . . . . 6 (Lim (rank‘𝐴) → (cf‘(rank‘𝐴)) ≼ {𝑤 ∣ ∃𝑥𝐴 𝑤 = (rank‘{𝑥})})
95 2fveq3 6886 . . . . . . . . . 10 (𝑦 = 𝐴 → (cf‘(rank‘𝑦)) = (cf‘(rank‘𝐴)))
96 breq12 5143 . . . . . . . . . 10 ((𝑦 = 𝐴 ∧ (cf‘(rank‘𝑦)) = (cf‘(rank‘𝐴))) → (𝑦 ≺ (cf‘(rank‘𝑦)) ↔ 𝐴 ≺ (cf‘(rank‘𝐴))))
9795, 96mpdan 684 . . . . . . . . 9 (𝑦 = 𝐴 → (𝑦 ≺ (cf‘(rank‘𝑦)) ↔ 𝐴 ≺ (cf‘(rank‘𝐴))))
98 rexeq 3313 . . . . . . . . . . 11 (𝑦 = 𝐴 → (∃𝑥𝑦 𝑤 = (rank‘{𝑥}) ↔ ∃𝑥𝐴 𝑤 = (rank‘{𝑥})))
9998abbidv 2793 . . . . . . . . . 10 (𝑦 = 𝐴 → {𝑤 ∣ ∃𝑥𝑦 𝑤 = (rank‘{𝑥})} = {𝑤 ∣ ∃𝑥𝐴 𝑤 = (rank‘{𝑥})})
100 breq12 5143 . . . . . . . . . 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 685 . . . . . . . . 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 2724 . . . . . . . . . 10 (𝑥𝑦 ↦ (rank‘{𝑥})) = (𝑥𝑦 ↦ (rank‘{𝑥}))
104103rnmpt 5944 . . . . . . . . 9 ran (𝑥𝑦 ↦ (rank‘{𝑥})) = {𝑤 ∣ ∃𝑥𝑦 𝑤 = (rank‘{𝑥})}
105 cfon 10246 . . . . . . . . . . 11 (cf‘(rank‘𝑦)) ∈ On
106 sdomdom 8972 . . . . . . . . . . 11 (𝑦 ≺ (cf‘(rank‘𝑦)) → 𝑦 ≼ (cf‘(rank‘𝑦)))
107 ondomen 10028 . . . . . . . . . . 11 (((cf‘(rank‘𝑦)) ∈ On ∧ 𝑦 ≼ (cf‘(rank‘𝑦))) → 𝑦 ∈ dom card)
108105, 106, 107sylancr 586 . . . . . . . . . 10 (𝑦 ≺ (cf‘(rank‘𝑦)) → 𝑦 ∈ dom card)
109 fvex 6894 . . . . . . . . . . . 12 (rank‘{𝑥}) ∈ V
110109, 103fnmpti 6683 . . . . . . . . . . 11 (𝑥𝑦 ↦ (rank‘{𝑥})) Fn 𝑦
111 dffn4 6801 . . . . . . . . . . 11 ((𝑥𝑦 ↦ (rank‘{𝑥})) Fn 𝑦 ↔ (𝑥𝑦 ↦ (rank‘{𝑥})):𝑦onto→ran (𝑥𝑦 ↦ (rank‘{𝑥})))
112110, 111mpbi 229 . . . . . . . . . 10 (𝑥𝑦 ↦ (rank‘{𝑥})):𝑦onto→ran (𝑥𝑦 ↦ (rank‘{𝑥}))
113 fodomnum 10048 . . . . . . . . . 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 5174 . . . . . . . 8 (𝑦 ≺ (cf‘(rank‘𝑦)) → {𝑤 ∣ ∃𝑥𝑦 𝑤 = (rank‘{𝑥})} ≼ 𝑦)
116102, 115vtoclg 3535 . . . . . . 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 8999 . . . . . . 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 681 . . . . 5 (Lim (rank‘𝐴) → (𝐴 ≺ (cf‘(rank‘𝐴)) → ¬ 𝐴 ≺ (cf‘(rank‘𝐴))))
121120pm2.01d 189 . . . 4 (Lim (rank‘𝐴) → ¬ 𝐴 ≺ (cf‘(rank‘𝐴)))
122121adantl 481 . . 3 (((rank‘𝐴) ∈ V ∧ Lim (rank‘𝐴)) → ¬ 𝐴 ≺ (cf‘(rank‘𝐴)))
1239, 41, 1223jaoi 1424 . 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 205  wa 395  w3o 1083   = wceq 1533  wcel 2098  {cab 2701  wne 2932  wrex 3062  Vcvv 3466  wss 3940  c0 4314  {csn 4620   cuni 4899   ciun 4987   class class class wbr 5138  cmpt 5221  dom cdm 5666  ran crn 5667  cima 5669  Oncon0 6354  Lim wlim 6355  suc csuc 6356   Fn wfn 6528  ontowfo 6531  cfv 6533  1oc1o 8454  cdom 8933  csdm 8934  𝑅1cr1 9753  rankcrnk 9754  cardccrd 9926  cfccf 9928
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-iin 4990  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-se 5622  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-r1 9755  df-rank 9756  df-card 9930  df-cf 9932  df-acn 9933
This theorem is referenced by:  inatsk  10769  grur1  10811
  Copyright terms: Public domain W3C validator