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Theorem sucidg 4881
Description: Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized). (Contributed by NM, 25-Mar-1995.) (Proof shortened by Scott Fenton, 20-Feb-2012.)
Assertion
Ref Expression
sucidg  |-  ( A  e.  V  ->  A  e.  suc  A )

Proof of Theorem sucidg
StepHypRef Expression
1 eqid 2450 . . 3  |-  A  =  A
21olci 391 . 2  |-  ( A  e.  A  \/  A  =  A )
3 elsucg 4870 . 2  |-  ( A  e.  V  ->  ( A  e.  suc  A  <->  ( A  e.  A  \/  A  =  A ) ) )
42, 3mpbiri 233 1  |-  ( A  e.  V  ->  A  e.  suc  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    = wceq 1370    e. wcel 1757   suc csuc 4805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-v 3056  df-un 3417  df-sn 3962  df-suc 4809
This theorem is referenced by:  sucid  4882  nsuceq0  4883  trsuc  4887  sucssel  4895  ordsuc  6511  onpsssuc  6516  nlimsucg  6539  tfrlem11  6933  tfrlem13  6935  tz7.44-2  6949  omeulem1  7107  oeordi  7112  oeeulem  7126  php4  7584  wofib  7846  suc11reg  7912  cantnfle  7966  cantnflt2  7968  cantnfp1lem3  7975  cantnflem1  7984  cantnfleOLD  7996  cantnflt2OLD  7998  cantnfp1lem3OLD  8001  cantnflem1OLD  8007  dfac12lem1  8399  dfac12lem2  8400  ttukeylem3  8767  ttukeylem7  8771  r1wunlim  8991  ontgval  28397
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