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Theorem opthpr 3611
Description: A way to represent ordered pairs using unordered pairs with distinct members. (Contributed by NM, 27-Mar-2007.)
Hypotheses
Ref Expression
preq12b.1  |-  A  e. 
_V
preq12b.2  |-  B  e. 
_V
preq12b.3  |-  C  e. 
_V
preq12b.4  |-  D  e. 
_V
Assertion
Ref Expression
opthpr  |-  ( A  =/=  D  ->  ( { A ,  B }  =  { C ,  D } 
<->  ( A  =  C  /\  B  =  D ) ) )

Proof of Theorem opthpr
StepHypRef Expression
1 preq12b.1 . . 3  |-  A  e. 
_V
2 preq12b.2 . . 3  |-  B  e. 
_V
3 preq12b.3 . . 3  |-  C  e. 
_V
4 preq12b.4 . . 3  |-  D  e. 
_V
51, 2, 3, 4preq12b 3609 . 2  |-  ( { A ,  B }  =  { C ,  D } 
<->  ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C )
) )
6 idd 21 . . . 4  |-  ( A  =/=  D  ->  (
( A  =  C  /\  B  =  D )  ->  ( A  =  C  /\  B  =  D ) ) )
7 df-ne 2256 . . . . . 6  |-  ( A  =/=  D  <->  -.  A  =  D )
8 pm2.21 582 . . . . . 6  |-  ( -.  A  =  D  -> 
( A  =  D  ->  ( B  =  C  ->  ( A  =  C  /\  B  =  D ) ) ) )
97, 8sylbi 119 . . . . 5  |-  ( A  =/=  D  ->  ( A  =  D  ->  ( B  =  C  -> 
( A  =  C  /\  B  =  D ) ) ) )
109impd 251 . . . 4  |-  ( A  =/=  D  ->  (
( A  =  D  /\  B  =  C )  ->  ( A  =  C  /\  B  =  D ) ) )
116, 10jaod 672 . . 3  |-  ( A  =/=  D  ->  (
( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C )
)  ->  ( A  =  C  /\  B  =  D ) ) )
12 orc 668 . . 3  |-  ( ( A  =  C  /\  B  =  D )  ->  ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C )
) )
1311, 12impbid1 140 . 2  |-  ( A  =/=  D  ->  (
( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C )
)  <->  ( A  =  C  /\  B  =  D ) ) )
145, 13syl5bb 190 1  |-  ( A  =/=  D  ->  ( { A ,  B }  =  { C ,  D } 
<->  ( A  =  C  /\  B  =  D ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664    = wceq 1289    e. wcel 1438    =/= wne 2255   _Vcvv 2619   {cpr 3442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-v 2621  df-un 3001  df-sn 3447  df-pr 3448
This theorem is referenced by: (None)
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