MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elsnc2g Unicode version

Theorem elsnc2g 3668
Description: There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that  B, rather than  A, be a set. (Contributed by NM, 28-Oct-2003.)
Assertion
Ref Expression
elsnc2g  |-  ( B  e.  V  ->  ( A  e.  { B } 
<->  A  =  B ) )

Proof of Theorem elsnc2g
StepHypRef Expression
1 elsni 3664 . 2  |-  ( A  e.  { B }  ->  A  =  B )
2 snidg 3665 . . 3  |-  ( B  e.  V  ->  B  e.  { B } )
3 eleq1 2343 . . 3  |-  ( A  =  B  ->  ( A  e.  { B } 
<->  B  e.  { B } ) )
42, 3syl5ibrcom 213 . 2  |-  ( B  e.  V  ->  ( A  =  B  ->  A  e.  { B }
) )
51, 4impbid2 195 1  |-  ( B  e.  V  ->  ( A  e.  { B } 
<->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   {csn 3640
This theorem is referenced by:  elsnc2  3669  elsuc2g  4460  mptiniseg  5167  fzosplitsni  10921  limcco  19243  ply1termlem  19585  stirlinglem8  27830  elpmapat  29953
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sn 3646
  Copyright terms: Public domain W3C validator