Theorem elsuci 4548

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 4516	. . . 4 |- suc B = ( B u. { B } )
2	1	eleq2i 2305	. . 3 |- ( A e. suc B <-> A e. ( B u. { B } ) )
3		elun 3370	. . 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 3727	. . 3 |- ( A e. { B } -> A = B )
6	5	orim2i 773	. 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 ) )

This proof depends on syntax axioms:   -> wi 4   \/ wo 720   = wceq 1402   e. wcel 2209   u. cun 3218   {csn 3709   suc csuc 4510

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3715  df-suc 4516

This theorem is used by:  trsucss  4568  onsucelsucexmid  4677  ordsoexmid  4709  ordsuc  4710  ordpwsucexmid  4717  nnsucelsuc  6764  nntri3or  6766  nnmordi  6789  nnaordex  6801  phplem3  7155  nninfninc  7463  nnnninf2  7467  3nelsucpw1  7593  3nsssucpw1  7595