Theorem sucidg 4441

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 2189	. . 3 |- A = A
2	1	olci 733	. 2 |- ( A e. A \/ A = A )
3		elsucg 4429	. 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 709   = wceq 1364   e. wcel 2160   suc csuc 4390

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2758  df-un 3152  df-sn 3620  df-suc 4396

This theorem is referenced by:  sucid  4442  nsuceq0g  4443  trsuc  4447  sucssel  4449  ordsucg  4526  sucunielr  4534  suc11g  4581  nlimsucg  4590  peano2b  4639  omsinds  4646  nnpredlt  4648  frecsuclem  6439  phplem4dom  6898  phplem4on  6903  dif1en  6915  fin0  6921  fin0or  6922  fidcenumlemrks  6991  bj-peano4  15246