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Theorem sucidg 4428
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 4417 . 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 4352
This theorem is referenced by:  sucid  4429  nsuceq0  4430  trsuc  4434  sucssel  4443  ordsuc  4563  onpsssuc  4568  nlimsucg  4591  tfrlem11  6358  tfrlem13  6360  tz7.44-2  6374  omeulem1  6534  oeordi  6539  oeeulem  6553  php4  7002  wofib  7214  suc11reg  7274  cantnfle  7326  cantnflt2  7328  cantnfp1lem3  7336  cantnflem1  7345  dfac12lem1  7723  dfac12lem2  7724  ttukeylem3  8092  ttukeylem7  8096  r1wunlim  8313  ontgval  24231
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 2759  df-un 3118  df-sn 3606  df-suc 4356
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