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Theorem preq12b 3974
Description: Equality relationship for 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
preq12b  |-  ( { A ,  B }  =  { C ,  D } 
<->  ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C )
) )

Proof of Theorem preq12b
StepHypRef Expression
1 preq12b.1 . . . . . 6  |-  A  e. 
_V
21prid1 3912 . . . . 5  |-  A  e. 
{ A ,  B }
3 eleq2 2497 . . . . 5  |-  ( { A ,  B }  =  { C ,  D }  ->  ( A  e. 
{ A ,  B } 
<->  A  e.  { C ,  D } ) )
42, 3mpbii 203 . . . 4  |-  ( { A ,  B }  =  { C ,  D }  ->  A  e.  { C ,  D }
)
51elpr 3832 . . . 4  |-  ( A  e.  { C ,  D }  <->  ( A  =  C  \/  A  =  D ) )
64, 5sylib 189 . . 3  |-  ( { A ,  B }  =  { C ,  D }  ->  ( A  =  C  \/  A  =  D ) )
7 preq1 3883 . . . . . . . 8  |-  ( A  =  C  ->  { A ,  B }  =  { C ,  B }
)
87eqeq1d 2444 . . . . . . 7  |-  ( A  =  C  ->  ( { A ,  B }  =  { C ,  D } 
<->  { C ,  B }  =  { C ,  D } ) )
9 preq12b.2 . . . . . . . 8  |-  B  e. 
_V
10 preq12b.4 . . . . . . . 8  |-  D  e. 
_V
119, 10preqr2 3973 . . . . . . 7  |-  ( { C ,  B }  =  { C ,  D }  ->  B  =  D )
128, 11syl6bi 220 . . . . . 6  |-  ( A  =  C  ->  ( { A ,  B }  =  { C ,  D }  ->  B  =  D ) )
1312com12 29 . . . . 5  |-  ( { A ,  B }  =  { C ,  D }  ->  ( A  =  C  ->  B  =  D ) )
1413ancld 537 . . . 4  |-  ( { A ,  B }  =  { C ,  D }  ->  ( A  =  C  ->  ( A  =  C  /\  B  =  D ) ) )
15 prcom 3882 . . . . . . 7  |-  { C ,  D }  =  { D ,  C }
1615eqeq2i 2446 . . . . . 6  |-  ( { A ,  B }  =  { C ,  D } 
<->  { A ,  B }  =  { D ,  C } )
17 preq1 3883 . . . . . . . . 9  |-  ( A  =  D  ->  { A ,  B }  =  { D ,  B }
)
1817eqeq1d 2444 . . . . . . . 8  |-  ( A  =  D  ->  ( { A ,  B }  =  { D ,  C } 
<->  { D ,  B }  =  { D ,  C } ) )
19 preq12b.3 . . . . . . . . 9  |-  C  e. 
_V
209, 19preqr2 3973 . . . . . . . 8  |-  ( { D ,  B }  =  { D ,  C }  ->  B  =  C )
2118, 20syl6bi 220 . . . . . . 7  |-  ( A  =  D  ->  ( { A ,  B }  =  { D ,  C }  ->  B  =  C ) )
2221com12 29 . . . . . 6  |-  ( { A ,  B }  =  { D ,  C }  ->  ( A  =  D  ->  B  =  C ) )
2316, 22sylbi 188 . . . . 5  |-  ( { A ,  B }  =  { C ,  D }  ->  ( A  =  D  ->  B  =  C ) )
2423ancld 537 . . . 4  |-  ( { A ,  B }  =  { C ,  D }  ->  ( A  =  D  ->  ( A  =  D  /\  B  =  C ) ) )
2514, 24orim12d 812 . . 3  |-  ( { A ,  B }  =  { C ,  D }  ->  ( ( A  =  C  \/  A  =  D )  ->  (
( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C ) ) ) )
266, 25mpd 15 . 2  |-  ( { A ,  B }  =  { C ,  D }  ->  ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C )
) )
27 preq12 3885 . . 3  |-  ( ( A  =  C  /\  B  =  D )  ->  { A ,  B }  =  { C ,  D } )
28 prcom 3882 . . . . 5  |-  { D ,  B }  =  { B ,  D }
2917, 28syl6eq 2484 . . . 4  |-  ( A  =  D  ->  { A ,  B }  =  { B ,  D }
)
30 preq1 3883 . . . 4  |-  ( B  =  C  ->  { B ,  D }  =  { C ,  D }
)
3129, 30sylan9eq 2488 . . 3  |-  ( ( A  =  D  /\  B  =  C )  ->  { A ,  B }  =  { C ,  D } )
3227, 31jaoi 369 . 2  |-  ( ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C ) )  ->  { A ,  B }  =  { C ,  D } )
3326, 32impbii 181 1  |-  ( { A ,  B }  =  { C ,  D } 
<->  ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956   {cpr 3815
This theorem is referenced by:  prel12  3975  opthpr  3976  preq12bg  3977  preqsn  3980  opeqpr  4453  preleq  7572  wlkdvspthlem  21607  altopthsn  25806  axlowdimlem13  25893
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-un 3325  df-sn 3820  df-pr 3821
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