Theorem elsuc2g 4322

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 4288	. . 3 |- suc B = ( B u. { B } )
2	1	eleq2i 2204	. 2 |- ( A e. suc B <-> A e. ( B u. { B } ) )
3		elun 3212	. . 3 |- ( A e. ( B u. { B } ) <-> ( A e. B \/ A e. { B } ) )
4		elsn2g 3553	. . . 4 |- ( B e. V -> ( A e. { B } <-> A = B ) )
5	4	orbi2d 779	. . 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 697   = wceq 1331   e. wcel 1480   u. cun 3064   {csn 3522   suc csuc 4282

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-sn 3528  df-suc 4288

This theorem is referenced by:  elsuc2  4324  nntri3or  6382  frec2uzltd  10169