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Theorem rankc1 9011
 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 9007 . . 3 (rank‘ 𝐴) ⊆ (rank‘𝐴)
32biantru 527 . 2 ((rank‘𝐴) ⊆ (rank‘ 𝐴) ↔ ((rank‘𝐴) ⊆ (rank‘ 𝐴) ∧ (rank‘ 𝐴) ⊆ (rank‘𝐴)))
4 iunss 4782 . . 3 ( 𝑥𝐴 suc (rank‘𝑥) ⊆ (rank‘ 𝐴) ↔ ∀𝑥𝐴 suc (rank‘𝑥) ⊆ (rank‘ 𝐴))
51rankval4 9008 . . . 4 (rank‘𝐴) = 𝑥𝐴 suc (rank‘𝑥)
65sseq1i 3855 . . 3 ((rank‘𝐴) ⊆ (rank‘ 𝐴) ↔ 𝑥𝐴 suc (rank‘𝑥) ⊆ (rank‘ 𝐴))
7 rankon 8936 . . . . 5 (rank‘𝑥) ∈ On
8 rankon 8936 . . . . 5 (rank‘ 𝐴) ∈ On
97, 8onsucssi 7303 . . . 4 ((rank‘𝑥) ∈ (rank‘ 𝐴) ↔ suc (rank‘𝑥) ⊆ (rank‘ 𝐴))
109ralbii 3190 . . 3 (∀𝑥𝐴 (rank‘𝑥) ∈ (rank‘ 𝐴) ↔ ∀𝑥𝐴 suc (rank‘𝑥) ⊆ (rank‘ 𝐴))
114, 6, 103bitr4ri 296 . 2 (∀𝑥𝐴 (rank‘𝑥) ∈ (rank‘ 𝐴) ↔ (rank‘𝐴) ⊆ (rank‘ 𝐴))
12 eqss 3843 . 2 ((rank‘𝐴) = (rank‘ 𝐴) ↔ ((rank‘𝐴) ⊆ (rank‘ 𝐴) ∧ (rank‘ 𝐴) ⊆ (rank‘𝐴)))
133, 11, 123bitr4i 295 1 (∀𝑥𝐴 (rank‘𝑥) ∈ (rank‘ 𝐴) ↔ (rank‘𝐴) = (rank‘ 𝐴))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   ∧ wa 386   = wceq 1658   ∈ wcel 2166  ∀wral 3118  Vcvv 3415   ⊆ wss 3799  ∪ cuni 4659  ∪ ciun 4741  suc csuc 5966  ‘cfv 6124  rankcrnk 8904 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210  ax-reg 8767  ax-inf2 8816 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-pss 3815  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4660  df-int 4699  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-tr 4977  df-id 5251  df-eprel 5256  df-po 5264  df-so 5265  df-fr 5302  df-we 5304  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-pred 5921  df-ord 5967  df-on 5968  df-lim 5969  df-suc 5970  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-om 7328  df-wrecs 7673  df-recs 7735  df-rdg 7773  df-r1 8905  df-rank 8906 This theorem is referenced by: (None)
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