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Theorem rankc2 9288
Description: A relationship that can be used for computation of rank. (Contributed by NM, 16-Sep-2006.)
Hypothesis
Ref Expression
rankr1b.1 𝐴 ∈ V
Assertion
Ref Expression
rankc2 (∃𝑥𝐴 (rank‘𝑥) = (rank‘ 𝐴) → (rank‘𝐴) = suc (rank‘ 𝐴))
Distinct variable group:   𝑥,𝐴

Proof of Theorem rankc2
StepHypRef Expression
1 pwuni 4840 . . . . 5 𝐴 ⊆ 𝒫 𝐴
2 rankr1b.1 . . . . . . . 8 𝐴 ∈ V
32uniex 7451 . . . . . . 7 𝐴 ∈ V
43pwex 5249 . . . . . 6 𝒫 𝐴 ∈ V
54rankss 9266 . . . . 5 (𝐴 ⊆ 𝒫 𝐴 → (rank‘𝐴) ⊆ (rank‘𝒫 𝐴))
61, 5ax-mp 5 . . . 4 (rank‘𝐴) ⊆ (rank‘𝒫 𝐴)
73rankpw 9260 . . . 4 (rank‘𝒫 𝐴) = suc (rank‘ 𝐴)
86, 7sseqtri 3954 . . 3 (rank‘𝐴) ⊆ suc (rank‘ 𝐴)
98a1i 11 . 2 (∃𝑥𝐴 (rank‘𝑥) = (rank‘ 𝐴) → (rank‘𝐴) ⊆ suc (rank‘ 𝐴))
102rankel 9256 . . . . 5 (𝑥𝐴 → (rank‘𝑥) ∈ (rank‘𝐴))
11 eleq1 2880 . . . . 5 ((rank‘𝑥) = (rank‘ 𝐴) → ((rank‘𝑥) ∈ (rank‘𝐴) ↔ (rank‘ 𝐴) ∈ (rank‘𝐴)))
1210, 11syl5ibcom 248 . . . 4 (𝑥𝐴 → ((rank‘𝑥) = (rank‘ 𝐴) → (rank‘ 𝐴) ∈ (rank‘𝐴)))
1312rexlimiv 3242 . . 3 (∃𝑥𝐴 (rank‘𝑥) = (rank‘ 𝐴) → (rank‘ 𝐴) ∈ (rank‘𝐴))
14 rankon 9212 . . . 4 (rank‘ 𝐴) ∈ On
15 rankon 9212 . . . 4 (rank‘𝐴) ∈ On
1614, 15onsucssi 7540 . . 3 ((rank‘ 𝐴) ∈ (rank‘𝐴) ↔ suc (rank‘ 𝐴) ⊆ (rank‘𝐴))
1713, 16sylib 221 . 2 (∃𝑥𝐴 (rank‘𝑥) = (rank‘ 𝐴) → suc (rank‘ 𝐴) ⊆ (rank‘𝐴))
189, 17eqssd 3935 1 (∃𝑥𝐴 (rank‘𝑥) = (rank‘ 𝐴) → (rank‘𝐴) = suc (rank‘ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2112  wrex 3110  Vcvv 3444  wss 3884  𝒫 cpw 4500   cuni 4803  suc csuc 6165  cfv 6328  rankcrnk 9180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-reg 9044  ax-inf2 9092
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-om 7565  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-r1 9181  df-rank 9182
This theorem is referenced by: (None)
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