Theorem elsuci 4140

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 4108	. . . 4 |- suc B = ( B u. { B } )
2	1	eleq2i 2104	. . 3 |- ( A e. suc B <-> A e. ( B u. { B } ) )
3		elun 3084	. . 3 |- ( A e. ( B u. { B } ) <-> ( A e. B \/ A e. { B } ) )
4	2, 3	bitri 173	. 2 |- ( A e. suc B <-> ( A e. B \/ A e. { B } ) )
5		elsni 3393	. . 3 |- ( A e. { B } -> A = B )
6	5	orim2i 678	. 2 |- ( ( A e. B \/ A e. { B } ) -> ( A e. B \/ A = B ) )
7	4, 6	sylbi 114	1 |- ( A e. suc B -> ( A e. B \/ A = B ) )

Syntax hints:   -> wi 4   \/ wo 629   = wceq 1243   e. wcel 1393   u. cun 2915   {csn 3375   suc csuc 4102

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-suc 4108

This theorem is referenced by:  trsucss  4160  onsucelsucexmid  4255  ordsoexmid  4286  ordsuc  4287  ordpwsucexmid  4294  nnsucelsuc  6070  nntri3or  6072  nnmordi  6089  nnaordex  6100  phplem3  6317