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Theorem sucexb 7807
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 7750 . 2 ((𝐴 ∈ V ∧ {𝐴} ∈ V) ↔ (𝐴 ∪ {𝐴}) ∈ V)
2 snex 5433 . . 3 {𝐴} ∈ V
32biantru 529 . 2 (𝐴 ∈ V ↔ (𝐴 ∈ V ∧ {𝐴} ∈ V))
4 df-suc 6375 . . 3 suc 𝐴 = (𝐴 ∪ {𝐴})
54eleq1i 2820 . 2 (suc 𝐴 ∈ V ↔ (𝐴 ∪ {𝐴}) ∈ V)
61, 3, 53bitr4i 303 1 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  wcel 2099  Vcvv 3471  cun 3945  {csn 4629  suc csuc 6371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-sn 4630  df-pr 4632  df-uni 4909  df-suc 6375
This theorem is referenced by:  sucexg  7808  onsucb  7820  ordsucelsuc  7825  oeordi  8608  suc11reg  9643  rankxpsuc  9906  isf32lem2  10378  limsucncmpi  35929
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