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

Theorem eldifvsn 4147
Description: A set is an element of the universal class excluding a singleton iff it is not the singleton element. (Contributed by AV, 7-Apr-2019.)
Assertion
Ref Expression
eldifvsn  |-  ( A  e.  V  ->  ( A  e.  ( _V  \  { B } )  <-> 
A  =/=  B ) )

Proof of Theorem eldifvsn
StepHypRef Expression
1 elex 3104 . . 3  |-  ( A  e.  V  ->  A  e.  _V )
21biantrurd 508 . 2  |-  ( A  e.  V  ->  ( A  =/=  B  <->  ( A  e.  _V  /\  A  =/= 
B ) ) )
3 eldifsn 4140 . 2  |-  ( A  e.  ( _V  \  { B } )  <->  ( A  e.  _V  /\  A  =/= 
B ) )
42, 3syl6rbbr 264 1  |-  ( A  e.  V  ->  ( A  e.  ( _V  \  { B } )  <-> 
A  =/=  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1804    =/= wne 2638   _Vcvv 3095    \ cdif 3458   {csn 4014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-v 3097  df-dif 3464  df-sn 4015
This theorem is referenced by:  cnvimadfsn  6912
  Copyright terms: Public domain W3C validator