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Theorem sucidg 4486
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 2296 . . 3  |-  A  =  A
21olci 380 . 2  |-  ( A  e.  A  \/  A  =  A )
3 elsucg 4475 . 2  |-  ( A  e.  V  ->  ( A  e.  suc  A  <->  ( A  e.  A  \/  A  =  A ) ) )
42, 3mpbiri 224 1  |-  ( A  e.  V  ->  A  e.  suc  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    = wceq 1632    e. wcel 1696   suc csuc 4410
This theorem is referenced by:  sucid  4487  nsuceq0  4488  trsuc  4492  sucssel  4501  ordsuc  4621  onpsssuc  4626  nlimsucg  4649  tfrlem11  6420  tfrlem13  6422  tz7.44-2  6436  omeulem1  6596  oeordi  6601  oeeulem  6615  php4  7064  wofib  7276  suc11reg  7336  cantnfle  7388  cantnflt2  7390  cantnfp1lem3  7398  cantnflem1  7407  dfac12lem1  7785  dfac12lem2  7786  ttukeylem3  8154  ttukeylem7  8158  r1wunlim  8375  ontgval  24942
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170  df-sn 3659  df-suc 4414
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