MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  prel12 Unicode version

Theorem prel12 3805
Description: Equality of two unordered pairs. (Contributed by NM, 17-Oct-1996.)
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
prel12  |-  ( -.  A  =  B  -> 
( { A ,  B }  =  { C ,  D }  <->  ( A  e.  { C ,  D }  /\  B  e.  { C ,  D } ) ) )

Proof of Theorem prel12
StepHypRef Expression
1 preq12b.1 . . . . 5  |-  A  e. 
_V
21prid1 3747 . . . 4  |-  A  e. 
{ A ,  B }
3 eleq2 2357 . . . 4  |-  ( { A ,  B }  =  { C ,  D }  ->  ( A  e. 
{ A ,  B } 
<->  A  e.  { C ,  D } ) )
42, 3mpbii 202 . . 3  |-  ( { A ,  B }  =  { C ,  D }  ->  A  e.  { C ,  D }
)
5 preq12b.2 . . . . 5  |-  B  e. 
_V
65prid2 3748 . . . 4  |-  B  e. 
{ A ,  B }
7 eleq2 2357 . . . 4  |-  ( { A ,  B }  =  { C ,  D }  ->  ( B  e. 
{ A ,  B } 
<->  B  e.  { C ,  D } ) )
86, 7mpbii 202 . . 3  |-  ( { A ,  B }  =  { C ,  D }  ->  B  e.  { C ,  D }
)
94, 8jca 518 . 2  |-  ( { A ,  B }  =  { C ,  D }  ->  ( A  e. 
{ C ,  D }  /\  B  e.  { C ,  D }
) )
101elpr 3671 . . . 4  |-  ( A  e.  { C ,  D }  <->  ( A  =  C  \/  A  =  D ) )
11 eqeq2 2305 . . . . . . . . . . . 12  |-  ( B  =  D  ->  ( A  =  B  <->  A  =  D ) )
1211notbid 285 . . . . . . . . . . 11  |-  ( B  =  D  ->  ( -.  A  =  B  <->  -.  A  =  D ) )
13 orel2 372 . . . . . . . . . . 11  |-  ( -.  A  =  D  -> 
( ( A  =  C  \/  A  =  D )  ->  A  =  C ) )
1412, 13syl6bi 219 . . . . . . . . . 10  |-  ( B  =  D  ->  ( -.  A  =  B  ->  ( ( A  =  C  \/  A  =  D )  ->  A  =  C ) ) )
1514com3l 75 . . . . . . . . 9  |-  ( -.  A  =  B  -> 
( ( A  =  C  \/  A  =  D )  ->  ( B  =  D  ->  A  =  C ) ) )
1615imp 418 . . . . . . . 8  |-  ( ( -.  A  =  B  /\  ( A  =  C  \/  A  =  D ) )  -> 
( B  =  D  ->  A  =  C ) )
1716ancrd 537 . . . . . . 7  |-  ( ( -.  A  =  B  /\  ( A  =  C  \/  A  =  D ) )  -> 
( B  =  D  ->  ( A  =  C  /\  B  =  D ) ) )
18 eqeq2 2305 . . . . . . . . . . . 12  |-  ( B  =  C  ->  ( A  =  B  <->  A  =  C ) )
1918notbid 285 . . . . . . . . . . 11  |-  ( B  =  C  ->  ( -.  A  =  B  <->  -.  A  =  C ) )
20 orel1 371 . . . . . . . . . . 11  |-  ( -.  A  =  C  -> 
( ( A  =  C  \/  A  =  D )  ->  A  =  D ) )
2119, 20syl6bi 219 . . . . . . . . . 10  |-  ( B  =  C  ->  ( -.  A  =  B  ->  ( ( A  =  C  \/  A  =  D )  ->  A  =  D ) ) )
2221com3l 75 . . . . . . . . 9  |-  ( -.  A  =  B  -> 
( ( A  =  C  \/  A  =  D )  ->  ( B  =  C  ->  A  =  D ) ) )
2322imp 418 . . . . . . . 8  |-  ( ( -.  A  =  B  /\  ( A  =  C  \/  A  =  D ) )  -> 
( B  =  C  ->  A  =  D ) )
2423ancrd 537 . . . . . . 7  |-  ( ( -.  A  =  B  /\  ( A  =  C  \/  A  =  D ) )  -> 
( B  =  C  ->  ( A  =  D  /\  B  =  C ) ) )
2517, 24orim12d 811 . . . . . 6  |-  ( ( -.  A  =  B  /\  ( A  =  C  \/  A  =  D ) )  -> 
( ( B  =  D  \/  B  =  C )  ->  (
( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C ) ) ) )
265elpr 3671 . . . . . . 7  |-  ( B  e.  { C ,  D }  <->  ( B  =  C  \/  B  =  D ) )
27 orcom 376 . . . . . . 7  |-  ( ( B  =  C  \/  B  =  D )  <->  ( B  =  D  \/  B  =  C )
)
2826, 27bitri 240 . . . . . 6  |-  ( B  e.  { C ,  D }  <->  ( B  =  D  \/  B  =  C ) )
29 preq12b.3 . . . . . . 7  |-  C  e. 
_V
30 preq12b.4 . . . . . . 7  |-  D  e. 
_V
311, 5, 29, 30preq12b 3804 . . . . . 6  |-  ( { A ,  B }  =  { C ,  D } 
<->  ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C )
) )
3225, 28, 313imtr4g 261 . . . . 5  |-  ( ( -.  A  =  B  /\  ( A  =  C  \/  A  =  D ) )  -> 
( B  e.  { C ,  D }  ->  { A ,  B }  =  { C ,  D } ) )
3332ex 423 . . . 4  |-  ( -.  A  =  B  -> 
( ( A  =  C  \/  A  =  D )  ->  ( B  e.  { C ,  D }  ->  { A ,  B }  =  { C ,  D }
) ) )
3410, 33syl5bi 208 . . 3  |-  ( -.  A  =  B  -> 
( A  e.  { C ,  D }  ->  ( B  e.  { C ,  D }  ->  { A ,  B }  =  { C ,  D } ) ) )
3534imp3a 420 . 2  |-  ( -.  A  =  B  -> 
( ( A  e. 
{ C ,  D }  /\  B  e.  { C ,  D }
)  ->  { A ,  B }  =  { C ,  D }
) )
369, 35impbid2 195 1  |-  ( -.  A  =  B  -> 
( { A ,  B }  =  { C ,  D }  <->  ( A  e.  { C ,  D }  /\  B  e.  { C ,  D } ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   {cpr 3654
This theorem is referenced by:  dfac2  7773
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170  df-sn 3659  df-pr 3660
  Copyright terms: Public domain W3C validator