MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rankc1 Structured version   Visualization version   GIF version

Theorem rankc1 9283
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 9279 . . 3 (rank‘ 𝐴) ⊆ (rank‘𝐴)
32biantru 533 . 2 ((rank‘𝐴) ⊆ (rank‘ 𝐴) ↔ ((rank‘𝐴) ⊆ (rank‘ 𝐴) ∧ (rank‘ 𝐴) ⊆ (rank‘𝐴)))
4 iunss 4950 . . 3 ( 𝑥𝐴 suc (rank‘𝑥) ⊆ (rank‘ 𝐴) ↔ ∀𝑥𝐴 suc (rank‘𝑥) ⊆ (rank‘ 𝐴))
51rankval4 9280 . . . 4 (rank‘𝐴) = 𝑥𝐴 suc (rank‘𝑥)
65sseq1i 3979 . . 3 ((rank‘𝐴) ⊆ (rank‘ 𝐴) ↔ 𝑥𝐴 suc (rank‘𝑥) ⊆ (rank‘ 𝐴))
7 rankon 9208 . . . . 5 (rank‘𝑥) ∈ On
8 rankon 9208 . . . . 5 (rank‘ 𝐴) ∈ On
97, 8onsucssi 7539 . . . 4 ((rank‘𝑥) ∈ (rank‘ 𝐴) ↔ suc (rank‘𝑥) ⊆ (rank‘ 𝐴))
109ralbii 3159 . . 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 3966 . 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 1538  wcel 2115  wral 3132  Vcvv 3479  wss 3918   cuni 4819   ciun 4900  suc csuc 6174  cfv 6336  rankcrnk 9176
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444  ax-reg 9040  ax-inf2 9088
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4820  df-int 4858  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-tr 5154  df-id 5441  df-eprel 5446  df-po 5455  df-so 5456  df-fr 5495  df-we 5497  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-om 7564  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-r1 9177  df-rank 9178
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator