MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elsuci Structured version   Unicode version

Theorem elsuci 4944
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
StepHypRef Expression
1 df-suc 4884 . . . 4  |-  suc  B  =  ( B  u.  { B } )
21eleq2i 2545 . . 3  |-  ( A  e.  suc  B  <->  A  e.  ( B  u.  { B } ) )
3 elun 3645 . . 3  |-  ( A  e.  ( B  u.  { B } )  <->  ( A  e.  B  \/  A  e.  { B } ) )
42, 3bitri 249 . 2  |-  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  e.  { B } ) )
5 elsni 4052 . . 3  |-  ( A  e.  { B }  ->  A  =  B )
65orim2i 518 . 2  |-  ( ( A  e.  B  \/  A  e.  { B } )  ->  ( A  e.  B  \/  A  =  B )
)
74, 6sylbi 195 1  |-  ( A  e.  suc  B  -> 
( A  e.  B  \/  A  =  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    = wceq 1379    e. wcel 1767    u. cun 3474   {csn 4027   suc csuc 4880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-un 3481  df-sn 4028  df-suc 4884
This theorem is referenced by:  trsucss  4963  ordnbtwn  4968  suc11  4981  tfrlem11  7057  omordi  7215  nnmordi  7280  phplem3  7698  pssnn  7738  r1sdom  8192  cfsuc  8637  axdc3lem2  8831  axdc3lem4  8833  indpi  9285  ontgval  29501  onsucconi  29507  suctrALT  32724  suctrALT2VD  32734  suctrALT2  32735  suctrALTcf  32820  suctrALTcfVD  32821  suctrALT3  32822  bnj563  32897  bnj964  33098
  Copyright terms: Public domain W3C validator