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Theorem dfpr2 3516
Description: Alternate definition of unordered pair. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)
Assertion
Ref Expression
dfpr2  |-  { A ,  B }  =  {
x  |  ( x  =  A  \/  x  =  B ) }
Distinct variable groups:    x, A    x, B

Proof of Theorem dfpr2
StepHypRef Expression
1 df-pr 3504 . 2  |-  { A ,  B }  =  ( { A }  u.  { B } )
2 elun 3187 . . . 4  |-  ( x  e.  ( { A }  u.  { B } )  <->  ( x  e.  { A }  \/  x  e.  { B } ) )
3 velsn 3514 . . . . 5  |-  ( x  e.  { A }  <->  x  =  A )
4 velsn 3514 . . . . 5  |-  ( x  e.  { B }  <->  x  =  B )
53, 4orbi12i 738 . . . 4  |-  ( ( x  e.  { A }  \/  x  e.  { B } )  <->  ( x  =  A  \/  x  =  B ) )
62, 5bitri 183 . . 3  |-  ( x  e.  ( { A }  u.  { B } )  <->  ( x  =  A  \/  x  =  B ) )
76abbi2i 2232 . 2  |-  ( { A }  u.  { B } )  =  {
x  |  ( x  =  A  \/  x  =  B ) }
81, 7eqtri 2138 1  |-  { A ,  B }  =  {
x  |  ( x  =  A  \/  x  =  B ) }
Colors of variables: wff set class
Syntax hints:    \/ wo 682    = wceq 1316    e. wcel 1465   {cab 2103    u. cun 3039   {csn 3497   {cpr 3498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504
This theorem is referenced by:  elprg  3517  nfpr  3543  pwsnss  3700  minmax  10969  xrminmax  11002
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