Theorem elsuci 4158

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 4126	. . . 4 |- suc B = ( B u. { B } )
2	1	eleq2i 2145	. . 3 |- ( A e. suc B <-> A e. ( B u. { B } ) )
3		elun 3113	. . 3 |- ( A e. ( B u. { B } ) <-> ( A e. B \/ A e. { B } ) )
4	2, 3	bitri 182	. 2 |- ( A e. suc B <-> ( A e. B \/ A e. { B } ) )
5		elsni 3416	. . 3 |- ( A e. { B } -> A = B )
6	5	orim2i 710	. 2 |- ( ( A e. B \/ A e. { B } ) -> ( A e. B \/ A = B ) )
7	4, 6	sylbi 119	1 |- ( A e. suc B -> ( A e. B \/ A = B ) )

Syntax hints:   -> wi 4   \/ wo 661   = wceq 1284   e. wcel 1433   u. cun 2971   {csn 3398   suc csuc 4120

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-sn 3404  df-suc 4126

This theorem is referenced by:  trsucss  4178  onsucelsucexmid  4273  ordsoexmid  4305  ordsuc  4306  ordpwsucexmid  4313  nnsucelsuc  6093  nntri3or  6095  nnmordi  6112  nnaordex  6123  phplem3  6340