Theorem elsuc2 4329

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 4327	. 2 |- ( A e. _V -> ( B e. suc A <-> ( B e. A \/ B = A ) ) )
3	1, 2	ax-mp 5	1 |- ( B e. suc A <-> ( B e. A \/ B = A ) )

Syntax hints:   <-> wb 104   \/ wo 697   = wceq 1331   e. wcel 1480   _Vcvv 2686   suc csuc 4287

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-sn 3533  df-suc 4293

This theorem is referenced by:  nnsucelsuc  6387