MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sucidg Unicode version

Theorem sucidg 4363
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 2253 . . 3  |-  A  =  A
21olci 382 . 2  |-  ( A  e.  A  \/  A  =  A )
3 elsucg 4352 . 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 4287
This theorem is referenced by:  sucid  4364  nsuceq0  4365  trsuc  4369  sucssel  4378  ordsuc  4496  onpsssuc  4501  nlimsucg  4524  tfrlem11  6290  tfrlem13  6292  tz7.44-2  6306  omeulem1  6466  oeordi  6471  oeeulem  6485  php4  6933  wofib  7144  suc11reg  7204  cantnfle  7256  cantnflt2  7258  cantnfp1lem3  7266  cantnflem1  7275  dfac12lem1  7653  dfac12lem2  7654  ttukeylem3  8022  ttukeylem7  8026  r1wunlim  8239  ontgval  24044
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 1926  ax-ext 2234
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 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-v 2729  df-un 3083  df-sn 3550  df-suc 4291
  Copyright terms: Public domain W3C validator