Theorem elsuci 4451

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 4419	. . . 4 |- suc B = ( B u. { B } )
2	1	eleq2i 2272	. . 3 |- ( A e. suc B <-> A e. ( B u. { B } ) )
3		elun 3314	. . 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 3651	. . 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 2176   u. cun 3164   {csn 3633   suc csuc 4413

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-sn 3639  df-suc 4419

This theorem is referenced by:  trsucss  4471  onsucelsucexmid  4579  ordsoexmid  4611  ordsuc  4612  ordpwsucexmid  4619  nnsucelsuc  6579  nntri3or  6581  nnmordi  6604  nnaordex  6616  phplem3  6953  nninfninc  7227  nnnninf2  7231  3nelsucpw1  7348  3nsssucpw1  7350