Theorem elsuci 4524

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 4492	. . . 4 |- suc B = ( B u. { B } )
2	1	eleq2i 2299	. . 3 |- ( A e. suc B <-> A e. ( B u. { B } ) )
3		elun 3360	. . 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 3707	. . 3 |- ( A e. { B } -> A = B )
6	5	orim2i 769	. 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 716   = wceq 1398   e. wcel 2203   u. cun 3209   {csn 3689   suc csuc 4486

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-sn 3695  df-suc 4492

This theorem is referenced by:  trsucss  4544  onsucelsucexmid  4652  ordsoexmid  4684  ordsuc  4685  ordpwsucexmid  4692  nnsucelsuc  6724  nntri3or  6726  nnmordi  6749  nnaordex  6761  phplem3  7108  nninfninc  7414  nnnninf2  7418  3nelsucpw1  7544  3nsssucpw1  7546