MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rankc2 Structured version   Visualization version   GIF version

Theorem rankc2 8772
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 4506 . . . . 5 𝐴 ⊆ 𝒫 𝐴
2 rankr1b.1 . . . . . . . 8 𝐴 ∈ V
32uniex 6995 . . . . . . 7 𝐴 ∈ V
43pwex 4878 . . . . . 6 𝒫 𝐴 ∈ V
54rankss 8750 . . . . 5 (𝐴 ⊆ 𝒫 𝐴 → (rank‘𝐴) ⊆ (rank‘𝒫 𝐴))
61, 5ax-mp 5 . . . 4 (rank‘𝐴) ⊆ (rank‘𝒫 𝐴)
73rankpw 8744 . . . 4 (rank‘𝒫 𝐴) = suc (rank‘ 𝐴)
86, 7sseqtri 3670 . . 3 (rank‘𝐴) ⊆ suc (rank‘ 𝐴)
98a1i 11 . 2 (∃𝑥𝐴 (rank‘𝑥) = (rank‘ 𝐴) → (rank‘𝐴) ⊆ suc (rank‘ 𝐴))
102rankel 8740 . . . . 5 (𝑥𝐴 → (rank‘𝑥) ∈ (rank‘𝐴))
11 eleq1 2718 . . . . 5 ((rank‘𝑥) = (rank‘ 𝐴) → ((rank‘𝑥) ∈ (rank‘𝐴) ↔ (rank‘ 𝐴) ∈ (rank‘𝐴)))
1210, 11syl5ibcom 235 . . . 4 (𝑥𝐴 → ((rank‘𝑥) = (rank‘ 𝐴) → (rank‘ 𝐴) ∈ (rank‘𝐴)))
1312rexlimiv 3056 . . 3 (∃𝑥𝐴 (rank‘𝑥) = (rank‘ 𝐴) → (rank‘ 𝐴) ∈ (rank‘𝐴))
14 rankon 8696 . . . 4 (rank‘ 𝐴) ∈ On
15 rankon 8696 . . . 4 (rank‘𝐴) ∈ On
1614, 15onsucssi 7083 . . 3 ((rank‘ 𝐴) ∈ (rank‘𝐴) ↔ suc (rank‘ 𝐴) ⊆ (rank‘𝐴))
1713, 16sylib 208 . 2 (∃𝑥𝐴 (rank‘𝑥) = (rank‘ 𝐴) → suc (rank‘ 𝐴) ⊆ (rank‘𝐴))
189, 17eqssd 3653 1 (∃𝑥𝐴 (rank‘𝑥) = (rank‘ 𝐴) → (rank‘𝐴) = suc (rank‘ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  wrex 2942  Vcvv 3231  wss 3607  𝒫 cpw 4191   cuni 4468  suc csuc 5763  cfv 5926  rankcrnk 8664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-reg 8538  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-r1 8665  df-rank 8666
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator