Theorem elsucg 4264

Description: Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-1995.)

Assertion

Ref	Expression
elsucg	|- ( A e. V -> ( A e. suc B <-> ( A e. B \/ A = B ) ) )

Proof of Theorem elsucg

Step	Hyp	Ref	Expression
1		df-suc 4231	. . . 4 |- suc B = ( B u. { B } )
2	1	eleq2i 2166	. . 3 |- ( A e. suc B <-> A e. ( B u. { B } ) )
3		elun 3164	. . 3 |- ( A e. ( B u. { B } ) <-> ( A e. B \/ A e. { B } ) )
4	2, 3	bitri 183	. 2 |- ( A e. suc B <-> ( A e. B \/ A e. { B } ) )
5		elsng 3489	. . 3 |- ( A e. V -> ( A e. { B } <-> A = B ) )
6	5	orbi2d 745	. 2 |- ( A e. V -> ( ( A e. B \/ A e. { B } ) <-> ( A e. B \/ A = B ) ) )
7	4, 6	syl5bb 191	1 |- ( A e. V -> ( A e. suc B <-> ( A e. B \/ A = B ) ) )

Syntax hints:   -> wi 4   <-> wb 104   \/ wo 670   = wceq 1299   e. wcel 1448   u. cun 3019   {csn 3474   suc csuc 4225

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 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-un 3025  df-sn 3480  df-suc 4231

This theorem is referenced by:  elsuc  4266  elelsuc  4269  sucidg  4276  onsucelsucr  4362  onsucsssucexmid  4380  suc11g  4410  nnsssuc  6328  nlt1pig  7050  bj-peano4  12738