Theorem elsuc2g 4160

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 4126	. . 3 |- suc B = ( B u. { B } )
2	1	eleq2i 2145	. 2 |- ( A e. suc B <-> A e. ( B u. { B } ) )
3		elun 3113	. . 3 |- ( A e. ( B u. { B } ) <-> ( A e. B \/ A e. { B } ) )
4		elsn2g 3427	. . . 4 |- ( B e. V -> ( A e. { B } <-> A = B ) )
5	4	orbi2d 736	. . 3 |- ( B e. V -> ( ( A e. B \/ A e. { B } ) <-> ( A e. B \/ A = B ) ) )
6	3, 5	syl5bb 190	. 2 |- ( B e. V -> ( A e. ( B u. { B } ) <-> ( A e. B \/ A = B ) ) )
7	2, 6	syl5bb 190	1 |- ( B e. V -> ( A e. suc B <-> ( A e. B \/ A = B ) ) )

Syntax hints:   -> wi 4   <-> wb 103   \/ wo 661   = wceq 1284   e. wcel 1433   u. cun 2971   {csn 3398   suc csuc 4120

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-sn 3404  df-suc 4126

This theorem is referenced by:  elsuc2  4162  nntri3or  6095  frec2uzltd  9405