Theorem elelsuc 4387

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 4382	. 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 1343   e. wcel 2136   suc csuc 4343

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-sn 3582  df-suc 4349

This theorem is referenced by:  suctr  4399  ordsuc  4540  nnaordex  6495  fiintim  6894  exmidfodomrlemr  7158  exmidfodomrlemrALT  7159  3nelsucpw1  7190  ennnfonelemex  12347