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Theorem sucidg 4407
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 2256 . . 3  |-  A  =  A
21olci 382 . 2  |-  ( A  e.  A  \/  A  =  A )
3 elsucg 4396 . 2  |-  ( A  e.  V  ->  ( A  e.  suc  A  <->  ( A  e.  A  \/  A  =  A ) ) )
42, 3mpbiri 226 1  |-  ( A  e.  V  ->  A  e.  suc  A )
Colors of variables: wff set class
Syntax hints:    -> wi 6    \/ wo 359    = wceq 1619    e. wcel 1621   suc csuc 4331
This theorem is referenced by:  sucid  4408  nsuceq0  4409  trsuc  4413  sucssel  4422  ordsuc  4542  onpsssuc  4547  nlimsucg  4570  tfrlem11  6337  tfrlem13  6339  tz7.44-2  6353  omeulem1  6513  oeordi  6518  oeeulem  6532  php4  6981  wofib  7193  suc11reg  7253  cantnfle  7305  cantnflt2  7307  cantnfp1lem3  7315  cantnflem1  7324  dfac12lem1  7702  dfac12lem2  7703  ttukeylem3  8071  ttukeylem7  8075  r1wunlim  8292  ontgval  24210
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-v 2742  df-un 3099  df-sn 3587  df-suc 4335
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