Theorem elsuc 4437

Description: Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-2003.)

Hypothesis

Ref	Expression
elsuc.1	|- A e. _V

Assertion

Ref	Expression
elsuc	|- ( A e. suc B <-> ( A e. B \/ A = B ) )

Proof of Theorem elsuc

Step	Hyp	Ref	Expression
1		elsuc.1	. 2 |- A e. _V
2		elsucg 4435	. 2 |- ( A e. _V -> ( A e. suc B <-> ( A e. B \/ A = B ) ) )
3	1, 2	ax-mp 5	1 |- ( A e. suc B <-> ( A e. B \/ A = B ) )

Syntax hints:   <-> wb 105   \/ wo 709   = wceq 1364   e. wcel 2164   _Vcvv 2760   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:  sucel  4441  suctr  4452  0elsucexmid  4597  tfrlemisucaccv  6378  tfr1onlemsucaccv  6394  tfrcllemsucaccv  6407