Theorem sucidg 4462

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 2204	. . 3 |- A = A
2	1	olci 733	. 2 |- ( A e. A \/ A = A )
3		elsucg 4450	. 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 709   = wceq 1372   e. wcel 2175   suc csuc 4411

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169  df-sn 3638  df-suc 4417

This theorem is referenced by:  sucid  4463  nsuceq0g  4464  trsuc  4468  sucssel  4470  ordsucg  4549  sucunielr  4557  suc11g  4604  nlimsucg  4613  peano2b  4662  omsinds  4669  nnpredlt  4671  frecsuclem  6491  phplem4dom  6958  phplem4on  6963  dif1en  6975  fin0  6981  fin0or  6982  fidcenumlemrks  7054  bj-peano4  15853