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Theorem opeqpr 4750
Description: Equivalence for an ordered pair equal to an unordered pair. (Contributed by NM, 3-Jun-2008.) (Avoid depending on this detail.)
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 2476 . 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 4218 . . 3  |-  <. A ,  B >.  =  { { A } ,  { A ,  B } }
54eqeq2i 2485 . 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 4694 . . 3  |-  { A }  e.  _V
9 prex 4695 . . 3  |-  { A ,  B }  e.  _V
106, 7, 8, 9preq12b 4208 . 2  |-  ( { C ,  D }  =  { { A } ,  { A ,  B } }  <->  ( ( C  =  { A }  /\  D  =  { A ,  B }
)  \/  ( C  =  { A ,  B }  /\  D  =  { A } ) ) )
111, 5, 103bitri 271 1  |-  ( <. A ,  B >.  =  { C ,  D } 
<->  ( ( C  =  { A }  /\  D  =  { A ,  B } )  \/  ( C  =  { A ,  B }  /\  D  =  { A } ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118   {csn 4033   {cpr 4035   <.cop 4039
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-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  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-ne 2664  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040
This theorem is referenced by:  relop  5159
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