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

Theorem opthpr 3918
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 3916 . 2  |-  ( { A ,  B }  =  { C ,  D } 
<->  ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C )
) )
6 idd 22 . . . 4  |-  ( A  =/=  D  ->  (
( A  =  C  /\  B  =  D )  ->  ( A  =  C  /\  B  =  D ) ) )
7 df-ne 2552 . . . . . 6  |-  ( A  =/=  D  <->  -.  A  =  D )
8 pm2.21 102 . . . . . 6  |-  ( -.  A  =  D  -> 
( A  =  D  ->  ( B  =  C  ->  ( A  =  C  /\  B  =  D ) ) ) )
97, 8sylbi 188 . . . . 5  |-  ( A  =/=  D  ->  ( A  =  D  ->  ( B  =  C  -> 
( A  =  C  /\  B  =  D ) ) ) )
109imp3a 421 . . . 4  |-  ( A  =/=  D  ->  (
( A  =  D  /\  B  =  C )  ->  ( A  =  C  /\  B  =  D ) ) )
116, 10jaod 370 . . 3  |-  ( A  =/=  D  ->  (
( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C )
)  ->  ( A  =  C  /\  B  =  D ) ) )
12 orc 375 . . 3  |-  ( ( A  =  C  /\  B  =  D )  ->  ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C )
) )
1311, 12impbid1 195 . 2  |-  ( A  =/=  D  ->  (
( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C )
)  <->  ( A  =  C  /\  B  =  D ) ) )
145, 13syl5bb 249 1  |-  ( A  =/=  D  ->  ( { A ,  B }  =  { C ,  D } 
<->  ( A  =  C  /\  B  =  D ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2550   _Vcvv 2899   {cpr 3758
This theorem is referenced by:  brdom7disj  8342  brdom6disj  8343
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-v 2901  df-un 3268  df-sn 3763  df-pr 3764
  Copyright terms: Public domain W3C validator