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Theorem rankc1 9372
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 9368 . . 3 (rank‘ 𝐴) ⊆ (rank‘𝐴)
32biantru 533 . 2 ((rank‘𝐴) ⊆ (rank‘ 𝐴) ↔ ((rank‘𝐴) ⊆ (rank‘ 𝐴) ∧ (rank‘ 𝐴) ⊆ (rank‘𝐴)))
4 iunss 4931 . . 3 ( 𝑥𝐴 suc (rank‘𝑥) ⊆ (rank‘ 𝐴) ↔ ∀𝑥𝐴 suc (rank‘𝑥) ⊆ (rank‘ 𝐴))
51rankval4 9369 . . . 4 (rank‘𝐴) = 𝑥𝐴 suc (rank‘𝑥)
65sseq1i 3905 . . 3 ((rank‘𝐴) ⊆ (rank‘ 𝐴) ↔ 𝑥𝐴 suc (rank‘𝑥) ⊆ (rank‘ 𝐴))
7 rankon 9297 . . . . 5 (rank‘𝑥) ∈ On
8 rankon 9297 . . . . 5 (rank‘ 𝐴) ∈ On
97, 8onsucssi 7575 . . . 4 ((rank‘𝑥) ∈ (rank‘ 𝐴) ↔ suc (rank‘𝑥) ⊆ (rank‘ 𝐴))
109ralbii 3080 . . 3 (∀𝑥𝐴 (rank‘𝑥) ∈ (rank‘ 𝐴) ↔ ∀𝑥𝐴 suc (rank‘𝑥) ⊆ (rank‘ 𝐴))
114, 6, 103bitr4ri 307 . 2 (∀𝑥𝐴 (rank‘𝑥) ∈ (rank‘ 𝐴) ↔ (rank‘𝐴) ⊆ (rank‘ 𝐴))
12 eqss 3892 . 2 ((rank‘𝐴) = (rank‘ 𝐴) ↔ ((rank‘𝐴) ⊆ (rank‘ 𝐴) ∧ (rank‘ 𝐴) ⊆ (rank‘𝐴)))
133, 11, 123bitr4i 306 1 (∀𝑥𝐴 (rank‘𝑥) ∈ (rank‘ 𝐴) ↔ (rank‘𝐴) = (rank‘ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1542  wcel 2114  wral 3053  Vcvv 3398  wss 3843   cuni 4796   ciun 4881  suc csuc 6174  cfv 6339  rankcrnk 9265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-reg 9129  ax-inf2 9177
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-om 7600  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-r1 9266  df-rank 9267
This theorem is referenced by: (None)
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