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Theorem sucexb 6901
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 6856 . 2 ((𝐴 ∈ V ∧ {𝐴} ∈ V) ↔ (𝐴 ∪ {𝐴}) ∈ V)
2 snex 4835 . . 3 {𝐴} ∈ V
32biantru 525 . 2 (𝐴 ∈ V ↔ (𝐴 ∈ V ∧ {𝐴} ∈ V))
4 df-suc 5646 . . 3 suc 𝐴 = (𝐴 ∪ {𝐴})
54eleq1i 2679 . 2 (suc 𝐴 ∈ V ↔ (𝐴 ∪ {𝐴}) ∈ V)
61, 3, 53bitr4i 291 1 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383  wcel 1977  Vcvv 3173  cun 3538  {csn 4125  suc csuc 5642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126  df-pr 4128  df-uni 4373  df-suc 5646
This theorem is referenced by:  sucexg  6902  sucelon  6909  ordsucelsuc  6914  oeordi  7554  suc11reg  8399  rankxpsuc  8628  isf32lem2  9059  limsucncmpi  31614
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