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Theorem ranksuc 8978
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 5949 . . 3 suc 𝐴 = (𝐴 ∪ {𝐴})
21fveq2i 6414 . 2 (rank‘suc 𝐴) = (rank‘(𝐴 ∪ {𝐴}))
3 rankr1b.1 . . . 4 𝐴 ∈ V
4 snex 5105 . . . 4 {𝐴} ∈ V
53, 4rankun 8969 . . 3 (rank‘(𝐴 ∪ {𝐴})) = ((rank‘𝐴) ∪ (rank‘{𝐴}))
63ranksn 8967 . . . . 5 (rank‘{𝐴}) = suc (rank‘𝐴)
76uneq2i 3970 . . . 4 ((rank‘𝐴) ∪ (rank‘{𝐴})) = ((rank‘𝐴) ∪ suc (rank‘𝐴))
8 sssucid 6021 . . . . 5 (rank‘𝐴) ⊆ suc (rank‘𝐴)
9 ssequn1 3989 . . . . 5 ((rank‘𝐴) ⊆ suc (rank‘𝐴) ↔ ((rank‘𝐴) ∪ suc (rank‘𝐴)) = suc (rank‘𝐴))
108, 9mpbi 221 . . . 4 ((rank‘𝐴) ∪ suc (rank‘𝐴)) = suc (rank‘𝐴)
117, 10eqtri 2835 . . 3 ((rank‘𝐴) ∪ (rank‘{𝐴})) = suc (rank‘𝐴)
125, 11eqtri 2835 . 2 (rank‘(𝐴 ∪ {𝐴})) = suc (rank‘𝐴)
132, 12eqtri 2835 1 (rank‘suc 𝐴) = suc (rank‘𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1637  wcel 2157  Vcvv 3398  cun 3774  wss 3776  {csn 4377  suc csuc 5945  cfv 6104  rankcrnk 8876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-rep 4971  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182  ax-reg 8739  ax-inf2 8788
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-ral 3108  df-rex 3109  df-reu 3110  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-pss 3792  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-tp 4382  df-op 4384  df-uni 4638  df-int 4677  df-iun 4721  df-br 4852  df-opab 4914  df-mpt 4931  df-tr 4954  df-id 5226  df-eprel 5231  df-po 5239  df-so 5240  df-fr 5277  df-we 5279  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-pred 5900  df-ord 5946  df-on 5947  df-lim 5948  df-suc 5949  df-iota 6067  df-fun 6106  df-fn 6107  df-f 6108  df-f1 6109  df-fo 6110  df-f1o 6111  df-fv 6112  df-om 7299  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-r1 8877  df-rank 8878
This theorem is referenced by: (None)
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