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Theorem rankc2 8678
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 4859 . . . . 5 𝐴 ⊆ 𝒫 𝐴
2 rankr1b.1 . . . . . . . 8 𝐴 ∈ V
32uniex 6906 . . . . . . 7 𝐴 ∈ V
43pwex 4808 . . . . . 6 𝒫 𝐴 ∈ V
54rankss 8656 . . . . 5 (𝐴 ⊆ 𝒫 𝐴 → (rank‘𝐴) ⊆ (rank‘𝒫 𝐴))
61, 5ax-mp 5 . . . 4 (rank‘𝐴) ⊆ (rank‘𝒫 𝐴)
73rankpw 8650 . . . 4 (rank‘𝒫 𝐴) = suc (rank‘ 𝐴)
86, 7sseqtri 3616 . . 3 (rank‘𝐴) ⊆ suc (rank‘ 𝐴)
98a1i 11 . 2 (∃𝑥𝐴 (rank‘𝑥) = (rank‘ 𝐴) → (rank‘𝐴) ⊆ suc (rank‘ 𝐴))
102rankel 8646 . . . . 5 (𝑥𝐴 → (rank‘𝑥) ∈ (rank‘𝐴))
11 eleq1 2686 . . . . 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 3020 . . 3 (∃𝑥𝐴 (rank‘𝑥) = (rank‘ 𝐴) → (rank‘ 𝐴) ∈ (rank‘𝐴))
14 rankon 8602 . . . 4 (rank‘ 𝐴) ∈ On
15 rankon 8602 . . . 4 (rank‘𝐴) ∈ On
1614, 15onsucssi 6988 . . 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 3600 1 (∃𝑥𝐴 (rank‘𝑥) = (rank‘ 𝐴) → (rank‘𝐴) = suc (rank‘ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  wrex 2908  Vcvv 3186  wss 3555  𝒫 cpw 4130   cuni 4402  suc csuc 5684  cfv 5847  rankcrnk 8570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-reg 8441  ax-inf2 8482
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-r1 8571  df-rank 8572
This theorem is referenced by: (None)
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