Theorem sucidg 4481

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 2207	. . 3 |- A = A
2	1	olci 734	. 2 |- ( A e. A \/ A = A )
3		elsucg 4469	. 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 710   = wceq 1373   e. wcel 2178   suc csuc 4430

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-sn 3649  df-suc 4436

This theorem is referenced by:  sucid  4482  nsuceq0g  4483  trsuc  4487  sucssel  4489  ordsucg  4568  sucunielr  4576  suc11g  4623  nlimsucg  4632  peano2b  4681  omsinds  4688  nnpredlt  4690  frecsuclem  6515  phplem4dom  6984  phplem4on  6990  dif1en  7002  fin0  7008  fin0or  7009  fidcenumlemrks  7081  bj-peano4  16090