Theorem elsuc2g 4383

Description: Variant of membership in a successor, requiring that B rather than A be a set. (Contributed by NM, 28-Oct-2003.)

Assertion

Ref	Expression
elsuc2g	|- ( B e. V -> ( A e. suc B <-> ( A e. B \/ A = B ) ) )

Proof of Theorem elsuc2g

Step	Hyp	Ref	Expression
1		df-suc 4349	. . 3 |- suc B = ( B u. { B } )
2	1	eleq2i 2233	. 2 |- ( A e. suc B <-> A e. ( B u. { B } ) )
3		elun 3263	. . 3 |- ( A e. ( B u. { B } ) <-> ( A e. B \/ A e. { B } ) )
4		elsn2g 3609	. . . 4 |- ( B e. V -> ( A e. { B } <-> A = B ) )
5	4	orbi2d 780	. . 3 |- ( B e. V -> ( ( A e. B \/ A e. { B } ) <-> ( A e. B \/ A = B ) ) )
6	3, 5	syl5bb 191	. 2 |- ( B e. V -> ( A e. ( B u. { B } ) <-> ( A e. B \/ A = B ) ) )
7	2, 6	syl5bb 191	1 |- ( B e. V -> ( A e. suc B <-> ( A e. B \/ A = B ) ) )

Syntax hints:   -> wi 4   <-> wb 104   \/ wo 698   = wceq 1343   e. wcel 2136   u. cun 3114   {csn 3576   suc csuc 4343

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-sn 3582  df-suc 4349

This theorem is referenced by:  elsuc2  4385  nntri3or  6461  frec2uzltd  10338