Theorem elsuci 4790

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 4730	. . . 4 |- suc B = ( B u. { B } )
2	1	eleq2i 2507	. . 3 |- ( A e. suc B <-> A e. ( B u. { B } ) )
3		elun 3502	. . 3 |- ( A e. ( B u. { B } ) <-> ( A e. B \/ A e. { B } ) )
4	2, 3	bitri 249	. 2 |- ( A e. suc B <-> ( A e. B \/ A e. { B } ) )
5		elsni 3907	. . 3 |- ( A e. { B } -> A = B )
6	5	orim2i 518	. 2 |- ( ( A e. B \/ A e. { B } ) -> ( A e. B \/ A = B ) )
7	4, 6	sylbi 195	1 |- ( A e. suc B -> ( A e. B \/ A = B ) )

Syntax hints:   -> wi 4   \/ wo 368   = wceq 1369   e. wcel 1756   u. cun 3331   {csn 3882   suc csuc 4726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-v 2979  df-un 3338  df-sn 3883  df-suc 4730

This theorem is referenced by:  trsucss  4809  ordnbtwn  4814  suc11  4827  tfrlem11  6852  omordi  7010  nnmordi  7075  phplem3  7497  pssnn  7536  r1sdom  7986  cfsuc  8431  axdc3lem2  8625  axdc3lem4  8627  indpi  9081  ontgval  28282  onsucconi  28288  suctrALT  31567  suctrALT2VD  31577  suctrALT2  31578  suctrALTcf  31663  suctrALTcfVD  31664  suctrALT3  31665  bnj563  31740  bnj964  31941