Theorem sucidg 4561

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 4549	. 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 )

This proof depends on syntax axioms:   -> wi 4   \/ wo 720   = wceq 1402   e. wcel 2209   suc csuc 4510

This proof depends on 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 proof 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 3715  df-suc 4516

This theorem is used by:  sucid  4562  nsuceq0g  4563  trsuc  4567  sucssel  4569  ordsucg  4649  sucunielr  4657  suc11g  4704  nlimsucg  4713  peano2b  4762  omsinds  4769  nnpredlt  4771  frecsuclem  6677  phplem4dom  7163  phplem4on  7169  dif1en  7183  fin0  7189  fin0or  7190  fidcenumlemrks  7270  bj-peano4  16981