Theorem elsuci 4325

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 4293	. . . 4 |- suc B = ( B u. { B } )
2	1	eleq2i 2206	. . 3 |- ( A e. suc B <-> A e. ( B u. { B } ) )
3		elun 3217	. . 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 3545	. . 3 |- ( A e. { B } -> A = B )
6	5	orim2i 750	. 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 697   = wceq 1331   e. wcel 1480   u. cun 3069   {csn 3527   suc csuc 4287

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-sn 3533  df-suc 4293

This theorem is referenced by:  trsucss  4345  onsucelsucexmid  4445  ordsoexmid  4477  ordsuc  4478  ordpwsucexmid  4485  nnsucelsuc  6387  nntri3or  6389  nnmordi  6412  nnaordex  6423  phplem3  6748