Theorem elsuc2g 4285

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 4251	. . 3 |- suc B = ( B u. { B } )
2	1	eleq2i 2179	. 2 |- ( A e. suc B <-> A e. ( B u. { B } ) )
3		elun 3181	. . 3 |- ( A e. ( B u. { B } ) <-> ( A e. B \/ A e. { B } ) )
4		elsn2g 3522	. . . 4 |- ( B e. V -> ( A e. { B } <-> A = B ) )
5	4	orbi2d 762	. . 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 680   = wceq 1312   e. wcel 1461   u. cun 3033   {csn 3491   suc csuc 4245

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 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-un 3039  df-sn 3497  df-suc 4251

This theorem is referenced by:  elsuc2  4287  nntri3or  6341  frec2uzltd  10063