Theorem elsuc 4391

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 4389	. 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 104   \/ wo 703   = wceq 1348   e. wcel 2141   _Vcvv 2730   suc csuc 4350

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3589  df-suc 4356

This theorem is referenced by:  sucel  4395  suctr  4406  0elsucexmid  4549  tfrlemisucaccv  6304  tfr1onlemsucaccv  6320  tfrcllemsucaccv  6333