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Theorem rankc1 8685
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
rankc1 (∀𝑥𝐴 (rank‘𝑥) ∈ (rank‘ 𝐴) ↔ (rank‘𝐴) = (rank‘ 𝐴))
Distinct variable group:   𝑥,𝐴

Proof of Theorem rankc1
StepHypRef Expression
1 rankr1b.1 . . . 4 𝐴 ∈ V
21rankuniss 8681 . . 3 (rank‘ 𝐴) ⊆ (rank‘𝐴)
32biantru 526 . 2 ((rank‘𝐴) ⊆ (rank‘ 𝐴) ↔ ((rank‘𝐴) ⊆ (rank‘ 𝐴) ∧ (rank‘ 𝐴) ⊆ (rank‘𝐴)))
4 iunss 4532 . . 3 ( 𝑥𝐴 suc (rank‘𝑥) ⊆ (rank‘ 𝐴) ↔ ∀𝑥𝐴 suc (rank‘𝑥) ⊆ (rank‘ 𝐴))
51rankval4 8682 . . . 4 (rank‘𝐴) = 𝑥𝐴 suc (rank‘𝑥)
65sseq1i 3613 . . 3 ((rank‘𝐴) ⊆ (rank‘ 𝐴) ↔ 𝑥𝐴 suc (rank‘𝑥) ⊆ (rank‘ 𝐴))
7 rankon 8610 . . . . 5 (rank‘𝑥) ∈ On
8 rankon 8610 . . . . 5 (rank‘ 𝐴) ∈ On
97, 8onsucssi 6995 . . . 4 ((rank‘𝑥) ∈ (rank‘ 𝐴) ↔ suc (rank‘𝑥) ⊆ (rank‘ 𝐴))
109ralbii 2975 . . 3 (∀𝑥𝐴 (rank‘𝑥) ∈ (rank‘ 𝐴) ↔ ∀𝑥𝐴 suc (rank‘𝑥) ⊆ (rank‘ 𝐴))
114, 6, 103bitr4ri 293 . 2 (∀𝑥𝐴 (rank‘𝑥) ∈ (rank‘ 𝐴) ↔ (rank‘𝐴) ⊆ (rank‘ 𝐴))
12 eqss 3602 . 2 ((rank‘𝐴) = (rank‘ 𝐴) ↔ ((rank‘𝐴) ⊆ (rank‘ 𝐴) ∧ (rank‘ 𝐴) ⊆ (rank‘𝐴)))
133, 11, 123bitr4i 292 1 (∀𝑥𝐴 (rank‘𝑥) ∈ (rank‘ 𝐴) ↔ (rank‘𝐴) = (rank‘ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  Vcvv 3189  wss 3559   cuni 4407   ciun 4490  suc csuc 5689  cfv 5852  rankcrnk 8578
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 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-reg 8449  ax-inf2 8490
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 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-om 7020  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-r1 8579  df-rank 8580
This theorem is referenced by: (None)
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