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Theorem sucexb 7523
Description: A successor exists iff its class argument exists. (Contributed by NM, 22-Jun-1998.)
Assertion
Ref Expression
sucexb (𝐴 ∈ V ↔ suc 𝐴 ∈ V)

Proof of Theorem sucexb
StepHypRef Expression
1 unexb 7470 . 2 ((𝐴 ∈ V ∧ {𝐴} ∈ V) ↔ (𝐴 ∪ {𝐴}) ∈ V)
2 snex 5331 . . 3 {𝐴} ∈ V
32biantru 532 . 2 (𝐴 ∈ V ↔ (𝐴 ∈ V ∧ {𝐴} ∈ V))
4 df-suc 6196 . . 3 suc 𝐴 = (𝐴 ∪ {𝐴})
54eleq1i 2903 . 2 (suc 𝐴 ∈ V ↔ (𝐴 ∪ {𝐴}) ∈ V)
61, 3, 53bitr4i 305 1 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398  wcel 2110  Vcvv 3494  cun 3933  {csn 4566  suc csuc 6192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-sn 4567  df-pr 4569  df-uni 4838  df-suc 6196
This theorem is referenced by:  sucexg  7524  sucelon  7531  ordsucelsuc  7536  oeordi  8212  suc11reg  9081  rankxpsuc  9310  isf32lem2  9775  limsucncmpi  33793
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