Theorem elsuci 4468

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 4436	. . . 4 |- suc B = ( B u. { B } )
2	1	eleq2i 2274	. . 3 |- ( A e. suc B <-> A e. ( B u. { B } ) )
3		elun 3322	. . 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 3661	. . 3 |- ( A e. { B } -> A = B )
6	5	orim2i 763	. 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 710   = wceq 1373   e. wcel 2178   u. cun 3172   {csn 3643   suc csuc 4430

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-sn 3649  df-suc 4436

This theorem is referenced by:  trsucss  4488  onsucelsucexmid  4596  ordsoexmid  4628  ordsuc  4629  ordpwsucexmid  4636  nnsucelsuc  6600  nntri3or  6602  nnmordi  6625  nnaordex  6637  phplem3  6976  nninfninc  7251  nnnninf2  7255  3nelsucpw1  7380  3nsssucpw1  7382