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Theorem preq12b 4191
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 4123 . . . . 5  |-  A  e. 
{ A ,  B }
3 eleq2 2516 . . . . 5  |-  ( { A ,  B }  =  { C ,  D }  ->  ( A  e. 
{ A ,  B } 
<->  A  e.  { C ,  D } ) )
42, 3mpbii 211 . . . 4  |-  ( { A ,  B }  =  { C ,  D }  ->  A  e.  { C ,  D }
)
51elpr 4032 . . . 4  |-  ( A  e.  { C ,  D }  <->  ( A  =  C  \/  A  =  D ) )
64, 5sylib 196 . . 3  |-  ( { A ,  B }  =  { C ,  D }  ->  ( A  =  C  \/  A  =  D ) )
7 preq1 4094 . . . . . . . 8  |-  ( A  =  C  ->  { A ,  B }  =  { C ,  B }
)
87eqeq1d 2445 . . . . . . 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 4190 . . . . . . 7  |-  ( { C ,  B }  =  { C ,  D }  ->  B  =  D )
128, 11syl6bi 228 . . . . . 6  |-  ( A  =  C  ->  ( { A ,  B }  =  { C ,  D }  ->  B  =  D ) )
1312com12 31 . . . . 5  |-  ( { A ,  B }  =  { C ,  D }  ->  ( A  =  C  ->  B  =  D ) )
1413ancld 553 . . . 4  |-  ( { A ,  B }  =  { C ,  D }  ->  ( A  =  C  ->  ( A  =  C  /\  B  =  D ) ) )
15 prcom 4093 . . . . . . 7  |-  { C ,  D }  =  { D ,  C }
1615eqeq2i 2461 . . . . . 6  |-  ( { A ,  B }  =  { C ,  D } 
<->  { A ,  B }  =  { D ,  C } )
17 preq1 4094 . . . . . . . . 9  |-  ( A  =  D  ->  { A ,  B }  =  { D ,  B }
)
1817eqeq1d 2445 . . . . . . . 8  |-  ( A  =  D  ->  ( { A ,  B }  =  { D ,  C } 
<->  { D ,  B }  =  { D ,  C } ) )
19 preq12b.3 . . . . . . . . 9  |-  C  e. 
_V
209, 19preqr2 4190 . . . . . . . 8  |-  ( { D ,  B }  =  { D ,  C }  ->  B  =  C )
2118, 20syl6bi 228 . . . . . . 7  |-  ( A  =  D  ->  ( { A ,  B }  =  { D ,  C }  ->  B  =  C ) )
2221com12 31 . . . . . 6  |-  ( { A ,  B }  =  { D ,  C }  ->  ( A  =  D  ->  B  =  C ) )
2316, 22sylbi 195 . . . . 5  |-  ( { A ,  B }  =  { C ,  D }  ->  ( A  =  D  ->  B  =  C ) )
2423ancld 553 . . . 4  |-  ( { A ,  B }  =  { C ,  D }  ->  ( A  =  D  ->  ( A  =  D  /\  B  =  C ) ) )
2514, 24orim12d 838 . . 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 4096 . . 3  |-  ( ( A  =  C  /\  B  =  D )  ->  { A ,  B }  =  { C ,  D } )
28 prcom 4093 . . . . 5  |-  { D ,  B }  =  { B ,  D }
2917, 28syl6eq 2500 . . . 4  |-  ( A  =  D  ->  { A ,  B }  =  { B ,  D }
)
30 preq1 4094 . . . 4  |-  ( B  =  C  ->  { B ,  D }  =  { C ,  D }
)
3129, 30sylan9eq 2504 . . 3  |-  ( ( A  =  D  /\  B  =  C )  ->  { A ,  B }  =  { C ,  D } )
3227, 31jaoi 379 . 2  |-  ( ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C ) )  ->  { A ,  B }  =  { C ,  D } )
3326, 32impbii 188 1  |-  ( { A ,  B }  =  { C ,  D } 
<->  ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1383    e. wcel 1804   _Vcvv 3095   {cpr 4016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-v 3097  df-un 3466  df-sn 4015  df-pr 4017
This theorem is referenced by:  prel12  4192  opthpr  4193  preq12bg  4194  preqsn  4198  opeqpr  4734  preleq  8037  axlowdimlem13  24129  wlkdvspthlem  24481  altopthsn  29586
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