Theorem elsuc 5794

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 5792	. 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 196   \/ wo 383   = wceq 1483   e. wcel 1990   _Vcvv 3200   suc csuc 5725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-sn 4178  df-suc 5729

This theorem is referenced by:  sucel  5798  suctrOLD  5809  limsssuc  7050  omsmolem  7733  cantnfle  8568  infxpenlem  8836  inatsk  9600  untsucf  31587  dfon2lem7  31694  nolesgn2ores  31825