Theorem elelsuc 4554

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 724	. 2 |- ( A e. B -> ( A e. B \/ A = B ) )
2		elsucg 4549	. 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 )

This proof depends on syntax axioms:   -> wi 4   \/ wo 720   = wceq 1402   e. wcel 2209   suc csuc 4510

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3715  df-suc 4516

This theorem is used by:  suctr  4566  ordsuc  4710  nnaordex  6801  fiintim  7238  exmidfodomrlemr  7554  exmidfodomrlemrALT  7555  3nelsucpw1  7593  ennnfonelemex  13305  3dom  17018