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Theorem sucid 4798
Description: A set belongs to its successor. (Contributed by NM, 22-Jun-1994.) (Proof shortened by Alan Sare, 18-Feb-2012.) (Proof shortened by Scott Fenton, 20-Feb-2012.)
Hypothesis
Ref Expression
sucid.1  |-  A  e. 
_V
Assertion
Ref Expression
sucid  |-  A  e. 
suc  A

Proof of Theorem sucid
StepHypRef Expression
1 sucid.1 . 2  |-  A  e. 
_V
2 sucidg 4797 . 2  |-  ( A  e.  _V  ->  A  e.  suc  A )
31, 2ax-mp 5 1  |-  A  e. 
suc  A
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1756   _Vcvv 2972   suc csuc 4721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-v 2974  df-un 3333  df-sn 3878  df-suc 4725
This theorem is referenced by:  eqelsuc  4800  unon  6442  onuninsuci  6451  tfinds  6470  peano5  6499  tfrlem16  6852  oawordeulem  6993  oalimcl  6999  omlimcl  7017  oneo  7020  omeulem1  7021  oeworde  7032  nnawordex  7076  nnneo  7090  phplem4  7493  php  7495  fiint  7588  inf0  7827  oancom  7857  cantnfval2  7877  cantnflt  7880  cantnflem1  7897  cantnfval2OLD  7907  cantnfltOLD  7910  cantnflem1OLD  7920  cnfcom  7933  cnfcom2  7935  cnfcom3lem  7936  cnfcom3  7937  cnfcomOLD  7941  cnfcom2OLD  7943  cnfcom3lemOLD  7944  cnfcom3OLD  7945  r1val1  7993  rankxplim3  8088  cardlim  8142  fseqenlem1  8194  cardaleph  8259  pwsdompw  8373  cfsmolem  8439  axdc3lem4  8622  ttukeylem5  8682  ttukeylem6  8683  ttukeylem7  8684  canthp1lem2  8820  pwxpndom2  8832  winainflem  8860  winalim2  8863  nqereu  9098  dfrdg2  27609  nofulllem5  27847  dford3lem2  29376  pw2f1ocnv  29386  aomclem1  29407  bnj216  31723  bnj98  31860
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