Theorem elelsuc 4440

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 713	. 2 |- ( A e. B -> ( A e. B \/ A = B ) )
2		elsucg 4435	. 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 709   = wceq 1364   e. wcel 2164   suc csuc 4396

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-sn 3624  df-suc 4402

This theorem is referenced by:  suctr  4452  ordsuc  4595  nnaordex  6581  fiintim  6985  exmidfodomrlemr  7262  exmidfodomrlemrALT  7263  3nelsucpw1  7294  ennnfonelemex  12571