Theorem elsuci 4494

Description: Membership in a successor. This one-way implication does not require that either A or B be sets. (Contributed by NM, 6-Jun-1994.)

Assertion

Ref	Expression
elsuci	|- ( A e. suc B -> ( A e. B \/ A = B ) )

Proof of Theorem elsuci

Step	Hyp	Ref	Expression
1		df-suc 4462	. . . 4 |- suc B = ( B u. { B } )
2	1	eleq2i 2296	. . 3 |- ( A e. suc B <-> A e. ( B u. { B } ) )
3		elun 3345	. . 3 |- ( A e. ( B u. { B } ) <-> ( A e. B \/ A e. { B } ) )
4	2, 3	bitri 184	. 2 |- ( A e. suc B <-> ( A e. B \/ A e. { B } ) )
5		elsni 3684	. . 3 |- ( A e. { B } -> A = B )
6	5	orim2i 766	. 2 |- ( ( A e. B \/ A e. { B } ) -> ( A e. B \/ A = B ) )
7	4, 6	sylbi 121	1 |- ( A e. suc B -> ( A e. B \/ A = B ) )

Syntax hints:   -> wi 4   \/ wo 713   = wceq 1395   e. wcel 2200   u. cun 3195   {csn 3666   suc csuc 4456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-sn 3672  df-suc 4462

This theorem is referenced by:  trsucss  4514  onsucelsucexmid  4622  ordsoexmid  4654  ordsuc  4655  ordpwsucexmid  4662  nnsucelsuc  6637  nntri3or  6639  nnmordi  6662  nnaordex  6674  phplem3  7015  nninfninc  7290  nnnninf2  7294  3nelsucpw1  7419  3nsssucpw1  7421