Theorem elelsuc 4331

Description: Membership in a successor. (Contributed by NM, 20-Jun-1998.)

Assertion

Ref	Expression
elelsuc	|- ( A e. B -> A e. suc B )

Proof of Theorem elelsuc

Step	Hyp	Ref	Expression
1		orc 701	. 2 |- ( A e. B -> ( A e. B \/ A = B ) )
2		elsucg 4326	. 2 |- ( A e. B -> ( A e. suc B <-> ( A e. B \/ A = B ) ) )
3	1, 2	mpbird 166	1 |- ( A e. B -> A e. suc B )

Syntax hints:   -> wi 4   \/ wo 697   = wceq 1331   e. wcel 1480   suc csuc 4287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-sn 3533  df-suc 4293

This theorem is referenced by:  suctr  4343  ordsuc  4478  nnaordex  6423  fiintim  6817  exmidfodomrlemr  7058  exmidfodomrlemrALT  7059  ennnfonelemex  11927