Theorem sucidg 4171

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 2081	. . 3 |- A = A
2	1	olci 683	. 2 |- ( A e. A \/ A = A )
3		elsucg 4159	. 2 |- ( A e. V -> ( A e. suc A <-> ( A e. A \/ A = A ) ) )
4	2, 3	mpbiri 166	1 |- ( A e. V -> A e. suc A )

Syntax hints:   -> wi 4   \/ wo 661   = wceq 1284   e. wcel 1433   suc csuc 4120

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-sn 3404  df-suc 4126

This theorem is referenced by:  sucid  4172  nsuceq0g  4173  trsuc  4177  sucssel  4179  ordsucg  4246  sucunielr  4254  suc11g  4300  nlimsucg  4309  peano2b  4355  frecsuclem2  6012  phplem4dom  6348  phplem4on  6353  dif1en  6364  fin0  6369  fin0or  6370  bj-peano4  10750