Theorem elsuc2g 4142

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 4108	. . 3 |- suc B = ( B u. { B } )
2	1	eleq2i 2104	. 2 |- ( A e. suc B <-> A e. ( B u. { B } ) )
3		elun 3084	. . 3 |- ( A e. ( B u. { B } ) <-> ( A e. B \/ A e. { B } ) )
4		elsn2g 3404	. . . 4 |- ( B e. V -> ( A e. { B } <-> A = B ) )
5	4	orbi2d 704	. . 3 |- ( B e. V -> ( ( A e. B \/ A e. { B } ) <-> ( A e. B \/ A = B ) ) )
6	3, 5	syl5bb 181	. 2 |- ( B e. V -> ( A e. ( B u. { B } ) <-> ( A e. B \/ A = B ) ) )
7	2, 6	syl5bb 181	1 |- ( B e. V -> ( A e. suc B <-> ( A e. B \/ A = B ) ) )

Syntax hints:   -> wi 4   <-> wb 98   \/ wo 629   = wceq 1243   e. wcel 1393   u. cun 2915   {csn 3375   suc csuc 4102

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-suc 4108

This theorem is referenced by:  elsuc2  4144  nntri3or  6072  frec2uzltd  9189