Theorem elelsuc 4532

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 720	. 2 |- ( A e. B -> ( A e. B \/ A = B ) )
2		elsucg 4527	. 2 |- ( A e. B -> ( A e. suc B <-> ( A e. B \/ A = B ) ) )
3	1, 2	mpbird 167	1 |- ( A e. B -> A e. suc B )

Syntax hints:   -> wi 4   \/ wo 716   = wceq 1398   e. wcel 2205   suc csuc 4488

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3217  df-sn 3697  df-suc 4494

This theorem is referenced by:  suctr  4544  ordsuc  4687  nnaordex  6763  fiintim  7193  exmidfodomrlemr  7507  exmidfodomrlemrALT  7508  3nelsucpw1  7546  ennnfonelemex  13182  3dom  16779