Theorem sucidg 4537

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

Step	Hyp	Ref	Expression
1		eqid 2232	. . 3 |- A = A
2	1	olci 740	. 2 |- ( A e. A \/ A = A )
3		elsucg 4525	. 2 |- ( A e. V -> ( A e. suc A <-> ( A e. A \/ A = A ) ) )
4	2, 3	mpbiri 168	1 |- ( A e. V -> A e. suc A )

Syntax hints:   -> wi 4   \/ wo 716   = wceq 1398   e. wcel 2203   suc csuc 4486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-sn 3695  df-suc 4492

This theorem is referenced by:  sucid  4538  nsuceq0g  4539  trsuc  4543  sucssel  4545  ordsucg  4624  sucunielr  4632  suc11g  4679  nlimsucg  4688  peano2b  4737  omsinds  4744  nnpredlt  4746  frecsuclem  6637  phplem4dom  7116  phplem4on  7122  dif1en  7136  fin0  7142  fin0or  7143  fidcenumlemrks  7223  bj-peano4  16725