Theorem elsuc 4546

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 4544	. 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 720   = wceq 1402   e. wcel 2209   _Vcvv 2821   suc csuc 4505

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3711  df-suc 4511

This theorem is referenced by:  sucel  4550  suctr  4561  0elsucexmid  4707  tfrlemisucaccv  6586  tfr1onlemsucaccv  6602  tfrcllemsucaccv  6615