Theorem elsuc2 4162

Description: Membership in a successor. (Contributed by NM, 15-Sep-2003.)

Hypothesis

Ref	Expression
elsuc.1	|- A e. _V

Assertion

Ref	Expression
elsuc2	|- ( B e. suc A <-> ( B e. A \/ B = A ) )

Proof of Theorem elsuc2

Step	Hyp	Ref	Expression
1		elsuc.1	. 2 |- A e. _V
2		elsuc2g 4160	. 2 |- ( A e. _V -> ( B e. suc A <-> ( B e. A \/ B = A ) ) )
3	1, 2	ax-mp 7	1 |- ( B e. suc A <-> ( B e. A \/ B = A ) )

Syntax hints:   <-> wb 103   \/ wo 661   = wceq 1284   e. wcel 1433   _Vcvv 2601   suc csuc 4120

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-sn 3404  df-suc 4126

This theorem is referenced by:  nnsucelsuc  6093