Theorem sucidg 4556

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 2238	. . 3 |- A = A
2	1	olci 744	. 2 |- ( A e. A \/ A = A )
3		elsucg 4544	. 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 720   = wceq 1402   e. wcel 2209   suc csuc 4505

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3711  df-suc 4511

This theorem is referenced by:  sucid  4557  nsuceq0g  4558  trsuc  4562  sucssel  4564  ordsucg  4644  sucunielr  4652  suc11g  4699  nlimsucg  4708  peano2b  4757  omsinds  4764  nnpredlt  4766  frecsuclem  6667  phplem4dom  7153  phplem4on  7159  dif1en  7173  fin0  7179  fin0or  7180  fidcenumlemrks  7260  bj-peano4  16895