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Theorem prel12 4164
Description: Equality of two unordered pairs. (Contributed by NM, 17-Oct-1996.)
Hypotheses
Ref Expression
preqr1.a  |-  A  e. 
_V
preqr1.b  |-  B  e. 
_V
preq12b.c  |-  C  e. 
_V
preq12b.d  |-  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 preqr1.a . . . . 5  |-  A  e. 
_V
21prid1 4092 . . . 4  |-  A  e. 
{ A ,  B }
3 eleq2 2528 . . . 4  |-  ( { A ,  B }  =  { C ,  D }  ->  ( A  e. 
{ A ,  B } 
<->  A  e.  { C ,  D } ) )
42, 3mpbii 216 . . 3  |-  ( { A ,  B }  =  { C ,  D }  ->  A  e.  { C ,  D }
)
5 preqr1.b . . . . 5  |-  B  e. 
_V
65prid2 4093 . . . 4  |-  B  e. 
{ A ,  B }
7 eleq2 2528 . . . 4  |-  ( { A ,  B }  =  { C ,  D }  ->  ( B  e. 
{ A ,  B } 
<->  B  e.  { C ,  D } ) )
86, 7mpbii 216 . . 3  |-  ( { A ,  B }  =  { C ,  D }  ->  B  e.  { C ,  D }
)
94, 8jca 539 . 2  |-  ( { A ,  B }  =  { C ,  D }  ->  ( A  e. 
{ C ,  D }  /\  B  e.  { C ,  D }
) )
101elpr 3997 . . . 4  |-  ( A  e.  { C ,  D }  <->  ( A  =  C  \/  A  =  D ) )
11 eqeq2 2472 . . . . . . . . . . . 12  |-  ( B  =  D  ->  ( A  =  B  <->  A  =  D ) )
1211notbid 300 . . . . . . . . . . 11  |-  ( B  =  D  ->  ( -.  A  =  B  <->  -.  A  =  D ) )
13 orel2 389 . . . . . . . . . . 11  |-  ( -.  A  =  D  -> 
( ( A  =  C  \/  A  =  D )  ->  A  =  C ) )
1412, 13syl6bi 236 . . . . . . . . . 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 435 . . . . . . . 8  |-  ( ( -.  A  =  B  /\  ( A  =  C  \/  A  =  D ) )  -> 
( B  =  D  ->  A  =  C ) )
1716ancrd 561 . . . . . . 7  |-  ( ( -.  A  =  B  /\  ( A  =  C  \/  A  =  D ) )  -> 
( B  =  D  ->  ( A  =  C  /\  B  =  D ) ) )
18 eqeq2 2472 . . . . . . . . . . . 12  |-  ( B  =  C  ->  ( A  =  B  <->  A  =  C ) )
1918notbid 300 . . . . . . . . . . 11  |-  ( B  =  C  ->  ( -.  A  =  B  <->  -.  A  =  C ) )
20 orel1 388 . . . . . . . . . . 11  |-  ( -.  A  =  C  -> 
( ( A  =  C  \/  A  =  D )  ->  A  =  D ) )
2119, 20syl6bi 236 . . . . . . . . . 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 435 . . . . . . . 8  |-  ( ( -.  A  =  B  /\  ( A  =  C  \/  A  =  D ) )  -> 
( B  =  C  ->  A  =  D ) )
2423ancrd 561 . . . . . . 7  |-  ( ( -.  A  =  B  /\  ( A  =  C  \/  A  =  D ) )  -> 
( B  =  C  ->  ( A  =  D  /\  B  =  C ) ) )
2517, 24orim12d 854 . . . . . 6  |-  ( ( -.  A  =  B  /\  ( A  =  C  \/  A  =  D ) )  -> 
( ( B  =  D  \/  B  =  C )  ->  (
( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C ) ) ) )
265elpr 3997 . . . . . . 7  |-  ( B  e.  { C ,  D }  <->  ( B  =  C  \/  B  =  D ) )
27 orcom 393 . . . . . . 7  |-  ( ( B  =  C  \/  B  =  D )  <->  ( B  =  D  \/  B  =  C )
)
2826, 27bitri 257 . . . . . 6  |-  ( B  e.  { C ,  D }  <->  ( B  =  D  \/  B  =  C ) )
29 preq12b.c . . . . . . 7  |-  C  e. 
_V
30 preq12b.d . . . . . . 7  |-  D  e. 
_V
311, 5, 29, 30preq12b 4163 . . . . . 6  |-  ( { A ,  B }  =  { C ,  D } 
<->  ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C )
) )
3225, 28, 313imtr4g 278 . . . . 5  |-  ( ( -.  A  =  B  /\  ( A  =  C  \/  A  =  D ) )  -> 
( B  e.  { C ,  D }  ->  { A ,  B }  =  { C ,  D } ) )
3332ex 440 . . . 4  |-  ( -.  A  =  B  -> 
( ( A  =  C  \/  A  =  D )  ->  ( B  e.  { C ,  D }  ->  { A ,  B }  =  { C ,  D }
) ) )
3410, 33syl5bi 225 . . 3  |-  ( -.  A  =  B  -> 
( A  e.  { C ,  D }  ->  ( B  e.  { C ,  D }  ->  { A ,  B }  =  { C ,  D } ) ) )
3534impd 437 . 2  |-  ( -.  A  =  B  -> 
( ( A  e. 
{ C ,  D }  /\  B  e.  { C ,  D }
)  ->  { A ,  B }  =  { C ,  D }
) )
369, 35impbid2 209 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 189    \/ wo 374    /\ wa 375    = wceq 1454    e. wcel 1897   _Vcvv 3056   {cpr 3981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-v 3058  df-un 3420  df-sn 3980  df-pr 3982
This theorem is referenced by:  prel12g  4168  dfac2  8586
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