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Theorem sucexb 4789
Description: A successor exists iff its class argument exists. (Contributed by NM, 22-Jun-1998.)
Assertion
Ref Expression
sucexb  |-  ( A  e.  _V  <->  suc  A  e. 
_V )

Proof of Theorem sucexb
StepHypRef Expression
1 unexb 4709 . 2  |-  ( ( A  e.  _V  /\  { A }  e.  _V ) 
<->  ( A  u.  { A } )  e.  _V )
2 snex 4405 . . 3  |-  { A }  e.  _V
32biantru 492 . 2  |-  ( A  e.  _V  <->  ( A  e.  _V  /\  { A }  e.  _V )
)
4 df-suc 4587 . . 3  |-  suc  A  =  ( A  u.  { A } )
54eleq1i 2499 . 2  |-  ( suc 
A  e.  _V  <->  ( A  u.  { A } )  e.  _V )
61, 3, 53bitr4i 269 1  |-  ( A  e.  _V  <->  suc  A  e. 
_V )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    e. wcel 1725   _Vcvv 2956    u. cun 3318   {csn 3814   suc csuc 4583
This theorem is referenced by:  sucexg  4790  sucelon  4797  ordsucelsuc  4802  oeordi  6830  suc11reg  7574  rankxpsuc  7806  isf32lem2  8234  limsucncmpi  26195
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rex 2711  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-sn 3820  df-pr 3821  df-uni 4016  df-suc 4587
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