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Theorem ranksuc 9280
 Description: The rank of a successor. (Contributed by NM, 18-Sep-2006.)
Hypothesis
Ref Expression
rankr1b.1 𝐴 ∈ V
Assertion
Ref Expression
ranksuc (rank‘suc 𝐴) = suc (rank‘𝐴)

Proof of Theorem ranksuc
StepHypRef Expression
1 df-suc 6165 . . 3 suc 𝐴 = (𝐴 ∪ {𝐴})
21fveq2i 6648 . 2 (rank‘suc 𝐴) = (rank‘(𝐴 ∪ {𝐴}))
3 rankr1b.1 . . . 4 𝐴 ∈ V
4 snex 5297 . . . 4 {𝐴} ∈ V
53, 4rankun 9271 . . 3 (rank‘(𝐴 ∪ {𝐴})) = ((rank‘𝐴) ∪ (rank‘{𝐴}))
63ranksn 9269 . . . . 5 (rank‘{𝐴}) = suc (rank‘𝐴)
76uneq2i 4087 . . . 4 ((rank‘𝐴) ∪ (rank‘{𝐴})) = ((rank‘𝐴) ∪ suc (rank‘𝐴))
8 sssucid 6236 . . . . 5 (rank‘𝐴) ⊆ suc (rank‘𝐴)
9 ssequn1 4107 . . . . 5 ((rank‘𝐴) ⊆ suc (rank‘𝐴) ↔ ((rank‘𝐴) ∪ suc (rank‘𝐴)) = suc (rank‘𝐴))
108, 9mpbi 233 . . . 4 ((rank‘𝐴) ∪ suc (rank‘𝐴)) = suc (rank‘𝐴)
117, 10eqtri 2821 . . 3 ((rank‘𝐴) ∪ (rank‘{𝐴})) = suc (rank‘𝐴)
125, 11eqtri 2821 . 2 (rank‘(𝐴 ∪ {𝐴})) = suc (rank‘𝐴)
132, 12eqtri 2821 1 (rank‘suc 𝐴) = suc (rank‘𝐴)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ∪ cun 3879   ⊆ wss 3881  {csn 4525  suc csuc 6161  ‘cfv 6324  rankcrnk 9178 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443  ax-reg 9042  ax-inf2 9090 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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-om 7563  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-r1 9179  df-rank 9180 This theorem is referenced by: (None)
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