MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elpr2 Structured version   Unicode version

Theorem elpr2 4035
Description: A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 14-Oct-2005.)
Hypotheses
Ref Expression
elpr2.1  |-  B  e. 
_V
elpr2.2  |-  C  e. 
_V
Assertion
Ref Expression
elpr2  |-  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) )

Proof of Theorem elpr2
StepHypRef Expression
1 elprg 4032 . . 3  |-  ( A  e.  { B ,  C }  ->  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) ) )
21ibi 241 . 2  |-  ( A  e.  { B ,  C }  ->  ( A  =  B  \/  A  =  C ) )
3 elpr2.1 . . . . . 6  |-  B  e. 
_V
4 eleq1 2526 . . . . . 6  |-  ( A  =  B  ->  ( A  e.  _V  <->  B  e.  _V ) )
53, 4mpbiri 233 . . . . 5  |-  ( A  =  B  ->  A  e.  _V )
6 elpr2.2 . . . . . 6  |-  C  e. 
_V
7 eleq1 2526 . . . . . 6  |-  ( A  =  C  ->  ( A  e.  _V  <->  C  e.  _V ) )
86, 7mpbiri 233 . . . . 5  |-  ( A  =  C  ->  A  e.  _V )
95, 8jaoi 377 . . . 4  |-  ( ( A  =  B  \/  A  =  C )  ->  A  e.  _V )
10 elprg 4032 . . . 4  |-  ( A  e.  _V  ->  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) ) )
119, 10syl 16 . . 3  |-  ( ( A  =  B  \/  A  =  C )  ->  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C )
) )
1211ibir 242 . 2  |-  ( ( A  =  B  \/  A  =  C )  ->  A  e.  { B ,  C } )
132, 12impbii 188 1  |-  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    \/ wo 366    = wceq 1398    e. wcel 1823   _Vcvv 3106   {cpr 4018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3108  df-un 3466  df-sn 4017  df-pr 4019
This theorem is referenced by:  elxr  11328  nofv  29657  bj-elopg  35003
  Copyright terms: Public domain W3C validator