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Theorem rankc1 9908
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 9904 . . 3 (rank‘ 𝐴) ⊆ (rank‘𝐴)
32biantru 529 . 2 ((rank‘𝐴) ⊆ (rank‘ 𝐴) ↔ ((rank‘𝐴) ⊆ (rank‘ 𝐴) ∧ (rank‘ 𝐴) ⊆ (rank‘𝐴)))
4 iunss 5050 . . 3 ( 𝑥𝐴 suc (rank‘𝑥) ⊆ (rank‘ 𝐴) ↔ ∀𝑥𝐴 suc (rank‘𝑥) ⊆ (rank‘ 𝐴))
51rankval4 9905 . . . 4 (rank‘𝐴) = 𝑥𝐴 suc (rank‘𝑥)
65sseq1i 4024 . . 3 ((rank‘𝐴) ⊆ (rank‘ 𝐴) ↔ 𝑥𝐴 suc (rank‘𝑥) ⊆ (rank‘ 𝐴))
7 rankon 9833 . . . . 5 (rank‘𝑥) ∈ On
8 rankon 9833 . . . . 5 (rank‘ 𝐴) ∈ On
97, 8onsucssi 7862 . . . 4 ((rank‘𝑥) ∈ (rank‘ 𝐴) ↔ suc (rank‘𝑥) ⊆ (rank‘ 𝐴))
109ralbii 3091 . . 3 (∀𝑥𝐴 (rank‘𝑥) ∈ (rank‘ 𝐴) ↔ ∀𝑥𝐴 suc (rank‘𝑥) ⊆ (rank‘ 𝐴))
114, 6, 103bitr4ri 304 . 2 (∀𝑥𝐴 (rank‘𝑥) ∈ (rank‘ 𝐴) ↔ (rank‘𝐴) ⊆ (rank‘ 𝐴))
12 eqss 4011 . 2 ((rank‘𝐴) = (rank‘ 𝐴) ↔ ((rank‘𝐴) ⊆ (rank‘ 𝐴) ∧ (rank‘ 𝐴) ⊆ (rank‘𝐴)))
133, 11, 123bitr4i 303 1 (∀𝑥𝐴 (rank‘𝑥) ∈ (rank‘ 𝐴) ↔ (rank‘𝐴) = (rank‘ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  wss 3963   cuni 4912   ciun 4996  suc csuc 6388  cfv 6563  rankcrnk 9801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-reg 9630  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-r1 9802  df-rank 9803
This theorem is referenced by: (None)
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