Theorem elelsuc 4369

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 702	. 2 |- ( A e. B -> ( A e. B \/ A = B ) )
2		elsucg 4364	. 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 698   = wceq 1335   e. wcel 2128   suc csuc 4325

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-sn 3566  df-suc 4331

This theorem is referenced by:  suctr  4381  ordsuc  4521  nnaordex  6471  fiintim  6870  exmidfodomrlemr  7131  exmidfodomrlemrALT  7132  3nelsucpw1  7163  ennnfonelemex  12126