Theorem elsuci 4333

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 4301	. . . 4 |- suc B = ( B u. { B } )
2	1	eleq2i 2207	. . 3 |- ( A e. suc B <-> A e. ( B u. { B } ) )
3		elun 3222	. . 3 |- ( A e. ( B u. { B } ) <-> ( A e. B \/ A e. { B } ) )
4	2, 3	bitri 183	. 2 |- ( A e. suc B <-> ( A e. B \/ A e. { B } ) )
5		elsni 3550	. . 3 |- ( A e. { B } -> A = B )
6	5	orim2i 751	. 2 |- ( ( A e. B \/ A e. { B } ) -> ( A e. B \/ A = B ) )
7	4, 6	sylbi 120	1 |- ( A e. suc B -> ( A e. B \/ A = B ) )

Syntax hints:   -> wi 4   \/ wo 698   = wceq 1332   e. wcel 1481   u. cun 3074   {csn 3532   suc csuc 4295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-sn 3538  df-suc 4301

This theorem is referenced by:  trsucss  4353  onsucelsucexmid  4453  ordsoexmid  4485  ordsuc  4486  ordpwsucexmid  4493  nnsucelsuc  6395  nntri3or  6397  nnmordi  6420  nnaordex  6431  phplem3  6756