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Theorem opthpr 3790
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 3788 . 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 2448 . . . . . 6  |-  ( A  =/=  D  <->  -.  A  =  D )
8 pm2.21 100 . . . . . 6  |-  ( -.  A  =  D  -> 
( A  =  D  ->  ( B  =  C  ->  ( A  =  C  /\  B  =  D ) ) ) )
97, 8sylbi 187 . . . . 5  |-  ( A  =/=  D  ->  ( A  =  D  ->  ( B  =  C  -> 
( A  =  C  /\  B  =  D ) ) ) )
109imp3a 420 . . . 4  |-  ( A  =/=  D  ->  (
( A  =  D  /\  B  =  C )  ->  ( A  =  C  /\  B  =  D ) ) )
116, 10jaod 369 . . 3  |-  ( A  =/=  D  ->  (
( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C )
)  ->  ( A  =  C  /\  B  =  D ) ) )
12 orc 374 . . 3  |-  ( ( A  =  C  /\  B  =  D )  ->  ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C )
) )
1311, 12impbid1 194 . 2  |-  ( A  =/=  D  ->  (
( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C )
)  <->  ( A  =  C  /\  B  =  D ) ) )
145, 13syl5bb 248 1  |-  ( A  =/=  D  ->  ( { A ,  B }  =  { C ,  D } 
<->  ( A  =  C  /\  B  =  D ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788   {cpr 3641
This theorem is referenced by:  brdom7disj  8156  brdom6disj  8157
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-un 3157  df-sn 3646  df-pr 3647
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