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Theorem sucexb 7788
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 7731 . 2 ((𝐴 ∈ V ∧ {𝐴} ∈ V) ↔ (𝐴 ∪ {𝐴}) ∈ V)
2 snex 5424 . . 3 {𝐴} ∈ V
32biantru 529 . 2 (𝐴 ∈ V ↔ (𝐴 ∈ V ∧ {𝐴} ∈ V))
4 df-suc 6363 . . 3 suc 𝐴 = (𝐴 ∪ {𝐴})
54eleq1i 2818 . 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 2098  Vcvv 3468  cun 3941  {csn 4623  suc csuc 6359
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-sn 4624  df-pr 4626  df-uni 4903  df-suc 6363
This theorem is referenced by:  sucexg  7789  onsucb  7801  ordsucelsuc  7806  oeordi  8585  suc11reg  9613  rankxpsuc  9876  isf32lem2  10348  limsucncmpi  35838
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