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Theorem prel12 4115
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 4046 . . . 4  |-  A  e. 
{ A ,  B }
3 eleq2 2490 . . . 4  |-  ( { A ,  B }  =  { C ,  D }  ->  ( A  e. 
{ A ,  B } 
<->  A  e.  { C ,  D } ) )
42, 3mpbii 214 . . 3  |-  ( { A ,  B }  =  { C ,  D }  ->  A  e.  { C ,  D }
)
5 preq12b.2 . . . . 5  |-  B  e. 
_V
65prid2 4047 . . . 4  |-  B  e. 
{ A ,  B }
7 eleq2 2490 . . . 4  |-  ( { A ,  B }  =  { C ,  D }  ->  ( B  e. 
{ A ,  B } 
<->  B  e.  { C ,  D } ) )
86, 7mpbii 214 . . 3  |-  ( { A ,  B }  =  { C ,  D }  ->  B  e.  { C ,  D }
)
94, 8jca 534 . 2  |-  ( { A ,  B }  =  { C ,  D }  ->  ( A  e. 
{ C ,  D }  /\  B  e.  { C ,  D }
) )
101elpr 3954 . . . 4  |-  ( A  e.  { C ,  D }  <->  ( A  =  C  \/  A  =  D ) )
11 eqeq2 2434 . . . . . . . . . . . 12  |-  ( B  =  D  ->  ( A  =  B  <->  A  =  D ) )
1211notbid 295 . . . . . . . . . . 11  |-  ( B  =  D  ->  ( -.  A  =  B  <->  -.  A  =  D ) )
13 orel2 384 . . . . . . . . . . 11  |-  ( -.  A  =  D  -> 
( ( A  =  C  \/  A  =  D )  ->  A  =  C ) )
1412, 13syl6bi 231 . . . . . . . . . 10  |-  ( B  =  D  ->  ( -.  A  =  B  ->  ( ( A  =  C  \/  A  =  D )  ->  A  =  C ) ) )
1514com3l 84 . . . . . . . . 9  |-  ( -.  A  =  B  -> 
( ( A  =  C  \/  A  =  D )  ->  ( B  =  D  ->  A  =  C ) ) )
1615imp 430 . . . . . . . 8  |-  ( ( -.  A  =  B  /\  ( A  =  C  \/  A  =  D ) )  -> 
( B  =  D  ->  A  =  C ) )
1716ancrd 556 . . . . . . 7  |-  ( ( -.  A  =  B  /\  ( A  =  C  \/  A  =  D ) )  -> 
( B  =  D  ->  ( A  =  C  /\  B  =  D ) ) )
18 eqeq2 2434 . . . . . . . . . . . 12  |-  ( B  =  C  ->  ( A  =  B  <->  A  =  C ) )
1918notbid 295 . . . . . . . . . . 11  |-  ( B  =  C  ->  ( -.  A  =  B  <->  -.  A  =  C ) )
20 orel1 383 . . . . . . . . . . 11  |-  ( -.  A  =  C  -> 
( ( A  =  C  \/  A  =  D )  ->  A  =  D ) )
2119, 20syl6bi 231 . . . . . . . . . 10  |-  ( B  =  C  ->  ( -.  A  =  B  ->  ( ( A  =  C  \/  A  =  D )  ->  A  =  D ) ) )
2221com3l 84 . . . . . . . . 9  |-  ( -.  A  =  B  -> 
( ( A  =  C  \/  A  =  D )  ->  ( B  =  C  ->  A  =  D ) ) )
2322imp 430 . . . . . . . 8  |-  ( ( -.  A  =  B  /\  ( A  =  C  \/  A  =  D ) )  -> 
( B  =  C  ->  A  =  D ) )
2423ancrd 556 . . . . . . 7  |-  ( ( -.  A  =  B  /\  ( A  =  C  \/  A  =  D ) )  -> 
( B  =  C  ->  ( A  =  D  /\  B  =  C ) ) )
2517, 24orim12d 846 . . . . . 6  |-  ( ( -.  A  =  B  /\  ( A  =  C  \/  A  =  D ) )  -> 
( ( B  =  D  \/  B  =  C )  ->  (
( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C ) ) ) )
265elpr 3954 . . . . . . 7  |-  ( B  e.  { C ,  D }  <->  ( B  =  C  \/  B  =  D ) )
27 orcom 388 . . . . . . 7  |-  ( ( B  =  C  \/  B  =  D )  <->  ( B  =  D  \/  B  =  C )
)
2826, 27bitri 252 . . . . . 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 4114 . . . . . 6  |-  ( { A ,  B }  =  { C ,  D } 
<->  ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C )
) )
3225, 28, 313imtr4g 273 . . . . 5  |-  ( ( -.  A  =  B  /\  ( A  =  C  \/  A  =  D ) )  -> 
( B  e.  { C ,  D }  ->  { A ,  B }  =  { C ,  D } ) )
3332ex 435 . . . 4  |-  ( -.  A  =  B  -> 
( ( A  =  C  \/  A  =  D )  ->  ( B  e.  { C ,  D }  ->  { A ,  B }  =  { C ,  D }
) ) )
3410, 33syl5bi 220 . . 3  |-  ( -.  A  =  B  -> 
( A  e.  { C ,  D }  ->  ( B  e.  { C ,  D }  ->  { A ,  B }  =  { C ,  D } ) ) )
3534impd 432 . 2  |-  ( -.  A  =  B  -> 
( ( A  e. 
{ C ,  D }  /\  B  e.  { C ,  D }
)  ->  { A ,  B }  =  { C ,  D }
) )
369, 35impbid2 207 1  |-  ( -.  A  =  B  -> 
( { A ,  B }  =  { C ,  D }  <->  ( A  e.  { C ,  D }  /\  B  e.  { C ,  D } ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1872   _Vcvv 3017   {cpr 3938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-v 3019  df-un 3379  df-sn 3937  df-pr 3939
This theorem is referenced by:  prel12g  4118  dfac2  8507
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