Theorem sucidg 4506

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 4494	. 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 4455

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 2801  df-un 3201  df-sn 3672  df-suc 4461

This theorem is referenced by:  sucid  4507  nsuceq0g  4508  trsuc  4512  sucssel  4514  ordsucg  4593  sucunielr  4601  suc11g  4648  nlimsucg  4657  peano2b  4706  omsinds  4713  nnpredlt  4715  frecsuclem  6550  phplem4dom  7019  phplem4on  7025  dif1en  7037  fin0  7043  fin0or  7044  fidcenumlemrks  7116  bj-peano4  16276