MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  preq12b Structured version   Unicode version

Theorem preq12b 4192
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 4124 . . . . 5  |-  A  e. 
{ A ,  B }
3 eleq2 2527 . . . . 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 4034 . . . 4  |-  ( A  e.  { C ,  D }  <->  ( A  =  C  \/  A  =  D ) )
64, 5sylib 196 . . 3  |-  ( { A ,  B }  =  { C ,  D }  ->  ( A  =  C  \/  A  =  D ) )
7 preq1 4095 . . . . . . . 8  |-  ( A  =  C  ->  { A ,  B }  =  { C ,  B }
)
87eqeq1d 2456 . . . . . . 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 4191 . . . . . . 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 551 . . . 4  |-  ( { A ,  B }  =  { C ,  D }  ->  ( A  =  C  ->  ( A  =  C  /\  B  =  D ) ) )
15 prcom 4094 . . . . . . 7  |-  { C ,  D }  =  { D ,  C }
1615eqeq2i 2472 . . . . . 6  |-  ( { A ,  B }  =  { C ,  D } 
<->  { A ,  B }  =  { D ,  C } )
17 preq1 4095 . . . . . . . . 9  |-  ( A  =  D  ->  { A ,  B }  =  { D ,  B }
)
1817eqeq1d 2456 . . . . . . . 8  |-  ( A  =  D  ->  ( { A ,  B }  =  { D ,  C } 
<->  { D ,  B }  =  { D ,  C } ) )
19 preq12b.3 . . . . . . . . 9  |-  C  e. 
_V
209, 19preqr2 4191 . . . . . . . 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 551 . . . 4  |-  ( { A ,  B }  =  { C ,  D }  ->  ( A  =  D  ->  ( A  =  D  /\  B  =  C ) ) )
2514, 24orim12d 836 . . 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 4097 . . 3  |-  ( ( A  =  C  /\  B  =  D )  ->  { A ,  B }  =  { C ,  D } )
28 prcom 4094 . . . . 5  |-  { D ,  B }  =  { B ,  D }
2917, 28syl6eq 2511 . . . 4  |-  ( A  =  D  ->  { A ,  B }  =  { B ,  D }
)
30 preq1 4095 . . . 4  |-  ( B  =  C  ->  { B ,  D }  =  { C ,  D }
)
3129, 30sylan9eq 2515 . . 3  |-  ( ( A  =  D  /\  B  =  C )  ->  { A ,  B }  =  { C ,  D } )
3227, 31jaoi 377 . 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 366    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106   {cpr 4018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3108  df-un 3466  df-sn 4017  df-pr 4019
This theorem is referenced by:  prel12  4193  opthpr  4194  preq12bg  4195  preqsn  4199  opeqpr  4733  preleq  8025  axlowdimlem13  24459  wlkdvspthlem  24811  altopthsn  29839
  Copyright terms: Public domain W3C validator