Theorem elsuc 4288

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 4286	. 2 |- ( A e. _V -> ( A e. suc B <-> ( A e. B \/ A = B ) ) )
3	1, 2	ax-mp 7	1 |- ( A e. suc B <-> ( A e. B \/ A = B ) )

Syntax hints:   <-> wb 104   \/ wo 680   = wceq 1314   e. wcel 1463   _Vcvv 2657   suc csuc 4247

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-un 3041  df-sn 3499  df-suc 4253

This theorem is referenced by:  sucel  4292  suctr  4303  0elsucexmid  4440  tfrlemisucaccv  6176  tfr1onlemsucaccv  6192  tfrcllemsucaccv  6205