Theorem sucidg 4463

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 2205	. . 3 |- A = A
2	1	olci 734	. 2 |- ( A e. A \/ A = A )
3		elsucg 4451	. 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 2176   suc csuc 4412

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-sn 3639  df-suc 4418

This theorem is referenced by:  sucid  4464  nsuceq0g  4465  trsuc  4469  sucssel  4471  ordsucg  4550  sucunielr  4558  suc11g  4605  nlimsucg  4614  peano2b  4663  omsinds  4670  nnpredlt  4672  frecsuclem  6492  phplem4dom  6959  phplem4on  6964  dif1en  6976  fin0  6982  fin0or  6983  fidcenumlemrks  7055  bj-peano4  15891