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Theorem dfpr2 3436
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 3424 . 2  |-  { A ,  B }  =  ( { A }  u.  { B } )
2 elun 3124 . . . 4  |-  ( x  e.  ( { A }  u.  { B } )  <->  ( x  e.  { A }  \/  x  e.  { B } ) )
3 velsn 3434 . . . . 5  |-  ( x  e.  { A }  <->  x  =  A )
4 velsn 3434 . . . . 5  |-  ( x  e.  { B }  <->  x  =  B )
53, 4orbi12i 714 . . . 4  |-  ( ( x  e.  { A }  \/  x  e.  { B } )  <->  ( x  =  A  \/  x  =  B ) )
62, 5bitri 182 . . 3  |-  ( x  e.  ( { A }  u.  { B } )  <->  ( x  =  A  \/  x  =  B ) )
76abbi2i 2197 . 2  |-  ( { A }  u.  { B } )  =  {
x  |  ( x  =  A  \/  x  =  B ) }
81, 7eqtri 2103 1  |-  { A ,  B }  =  {
x  |  ( x  =  A  \/  x  =  B ) }
Colors of variables: wff set class
Syntax hints:    \/ wo 662    = wceq 1285    e. wcel 1434   {cab 2069    u. cun 2981   {csn 3417   {cpr 3418
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2987  df-sn 3423  df-pr 3424
This theorem is referenced by:  elprg  3437  nfpr  3461  pwsnss  3616  minmax  10297
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