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

Theorem opeqpr 4263
Description: Equivalence for an ordered pair equal to an unordered pair. (Contributed by NM, 3-Jun-2008.)
Hypotheses
Ref Expression
opeqpr.1  |-  A  e. 
_V
opeqpr.2  |-  B  e. 
_V
opeqpr.3  |-  C  e. 
_V
opeqpr.4  |-  D  e. 
_V
Assertion
Ref Expression
opeqpr  |-  ( <. A ,  B >.  =  { C ,  D } 
<->  ( ( C  =  { A }  /\  D  =  { A ,  B } )  \/  ( C  =  { A ,  B }  /\  D  =  { A } ) ) )

Proof of Theorem opeqpr
StepHypRef Expression
1 eqcom 2285 . 2  |-  ( <. A ,  B >.  =  { C ,  D } 
<->  { C ,  D }  =  <. A ,  B >. )
2 opeqpr.1 . . . 4  |-  A  e. 
_V
3 opeqpr.2 . . . 4  |-  B  e. 
_V
42, 3dfop 3795 . . 3  |-  <. A ,  B >.  =  { { A } ,  { A ,  B } }
54eqeq2i 2293 . 2  |-  ( { C ,  D }  =  <. A ,  B >.  <->  { C ,  D }  =  { { A } ,  { A ,  B } } )
6 opeqpr.3 . . 3  |-  C  e. 
_V
7 opeqpr.4 . . 3  |-  D  e. 
_V
8 snex 4216 . . 3  |-  { A }  e.  _V
9 prex 4217 . . 3  |-  { A ,  B }  e.  _V
106, 7, 8, 9preq12b 3788 . 2  |-  ( { C ,  D }  =  { { A } ,  { A ,  B } }  <->  ( ( C  =  { A }  /\  D  =  { A ,  B }
)  \/  ( C  =  { A ,  B }  /\  D  =  { A } ) ) )
111, 5, 103bitri 262 1  |-  ( <. A ,  B >.  =  { C ,  D } 
<->  ( ( C  =  { A }  /\  D  =  { A ,  B } )  \/  ( C  =  { A ,  B }  /\  D  =  { A } ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   {csn 3640   {cpr 3641   <.cop 3643
This theorem is referenced by:  relop  4834
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  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-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649
  Copyright terms: Public domain W3C validator