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Theorem dfpr2 3959
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 3947 . 2  |-  { A ,  B }  =  ( { A }  u.  { B } )
2 elun 3559 . . . 4  |-  ( x  e.  ( { A }  u.  { B } )  <->  ( x  e.  { A }  \/  x  e.  { B } ) )
3 elsn 3958 . . . . 5  |-  ( x  e.  { A }  <->  x  =  A )
4 elsn 3958 . . . . 5  |-  ( x  e.  { B }  <->  x  =  B )
53, 4orbi12i 519 . . . 4  |-  ( ( x  e.  { A }  \/  x  e.  { B } )  <->  ( x  =  A  \/  x  =  B ) )
62, 5bitri 249 . . 3  |-  ( x  e.  ( { A }  u.  { B } )  <->  ( x  =  A  \/  x  =  B ) )
76abbi2i 2515 . 2  |-  ( { A }  u.  { B } )  =  {
x  |  ( x  =  A  \/  x  =  B ) }
81, 7eqtri 2411 1  |-  { A ,  B }  =  {
x  |  ( x  =  A  \/  x  =  B ) }
Colors of variables: wff setvar class
Syntax hints:    \/ wo 366    = wceq 1399    e. wcel 1826   {cab 2367    u. cun 3387   {csn 3944   {cpr 3946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-v 3036  df-un 3394  df-sn 3945  df-pr 3947
This theorem is referenced by:  elprg  3960  nfpr  3991  pwpw0  4092  pwsn  4157  pwsnALT  4158  zfpair  4599  grothprimlem  9122  nb3graprlem1  24572
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