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Theorem dfpr2 3656
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 3647 . 2  |-  { A ,  B }  =  ( { A }  u.  { B } )
2 elun 3316 . . . 4  |-  ( x  e.  ( { A }  u.  { B } )  <->  ( x  e.  { A }  \/  x  e.  { B } ) )
3 elsn 3655 . . . . 5  |-  ( x  e.  { A }  <->  x  =  A )
4 elsn 3655 . . . . 5  |-  ( x  e.  { B }  <->  x  =  B )
53, 4orbi12i 507 . . . 4  |-  ( ( x  e.  { A }  \/  x  e.  { B } )  <->  ( x  =  A  \/  x  =  B ) )
62, 5bitri 240 . . 3  |-  ( x  e.  ( { A }  u.  { B } )  <->  ( x  =  A  \/  x  =  B ) )
76abbi2i 2394 . 2  |-  ( { A }  u.  { B } )  =  {
x  |  ( x  =  A  \/  x  =  B ) }
81, 7eqtri 2303 1  |-  { A ,  B }  =  {
x  |  ( x  =  A  \/  x  =  B ) }
Colors of variables: wff set class
Syntax hints:    \/ wo 357    = wceq 1623    e. wcel 1684   {cab 2269    u. cun 3150   {csn 3640   {cpr 3641
This theorem is referenced by:  elprg  3657  nfpr  3680  pwpw0  3763  pwsn  3821  pwsnALT  3822  zfpair  4212  grothprimlem  8455
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-un 3157  df-sn 3646  df-pr 3647
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