Theorem sucidg 4511

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 2229	. . 3 |- A = A
2	1	olci 737	. 2 |- ( A e. A \/ A = A )
3		elsucg 4499	. 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 713   = wceq 1395   e. wcel 2200   suc csuc 4460

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-un 3202  df-sn 3673  df-suc 4466

This theorem is referenced by:  sucid  4512  nsuceq0g  4513  trsuc  4517  sucssel  4519  ordsucg  4598  sucunielr  4606  suc11g  4653  nlimsucg  4662  peano2b  4711  omsinds  4718  nnpredlt  4720  frecsuclem  6567  phplem4dom  7043  phplem4on  7049  dif1en  7061  fin0  7067  fin0or  7068  fidcenumlemrks  7143  bj-peano4  16486