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Theorem opthpr 3706
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 3704 . 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 2310 . . . . . 6  |-  ( A  =/=  D  <->  -.  A  =  D )
8 pm2.21 607 . . . . . 6  |-  ( -.  A  =  D  -> 
( A  =  D  ->  ( B  =  C  ->  ( A  =  C  /\  B  =  D ) ) ) )
97, 8sylbi 120 . . . . 5  |-  ( A  =/=  D  ->  ( A  =  D  ->  ( B  =  C  -> 
( A  =  C  /\  B  =  D ) ) ) )
109impd 252 . . . 4  |-  ( A  =/=  D  ->  (
( A  =  D  /\  B  =  C )  ->  ( A  =  C  /\  B  =  D ) ) )
116, 10jaod 707 . . 3  |-  ( A  =/=  D  ->  (
( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C )
)  ->  ( A  =  C  /\  B  =  D ) ) )
12 orc 702 . . 3  |-  ( ( A  =  C  /\  B  =  D )  ->  ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C )
) )
1311, 12impbid1 141 . 2  |-  ( A  =/=  D  ->  (
( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C )
)  <->  ( A  =  C  /\  B  =  D ) ) )
145, 13syl5bb 191 1  |-  ( A  =/=  D  ->  ( { A ,  B }  =  { C ,  D } 
<->  ( A  =  C  /\  B  =  D ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 698    = wceq 1332    e. wcel 1481    =/= wne 2309   _Vcvv 2689   {cpr 3532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-v 2691  df-un 3079  df-sn 3537  df-pr 3538
This theorem is referenced by: (None)
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