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Theorem sucexb 7126
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 7075 . 2 ((𝐴 ∈ V ∧ {𝐴} ∈ V) ↔ (𝐴 ∪ {𝐴}) ∈ V)
2 snex 5013 . . 3 {𝐴} ∈ V
32biantru 527 . 2 (𝐴 ∈ V ↔ (𝐴 ∈ V ∧ {𝐴} ∈ V))
4 df-suc 5842 . . 3 suc 𝐴 = (𝐴 ∪ {𝐴})
54eleq1i 2794 . 2 (suc 𝐴 ∈ V ↔ (𝐴 ∪ {𝐴}) ∈ V)
61, 3, 53bitr4i 292 1 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  wcel 2103  Vcvv 3304  cun 3678  {csn 4285  suc csuc 5838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-rex 3020  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-sn 4286  df-pr 4288  df-uni 4545  df-suc 5842
This theorem is referenced by:  sucexg  7127  sucelon  7134  ordsucelsuc  7139  oeordi  7787  suc11reg  8629  rankxpsuc  8858  isf32lem2  9289  limsucncmpi  32671
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