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Theorem dfpr2 3995
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 3983 . 2  |-  { A ,  B }  =  ( { A }  u.  { B } )
2 elun 3586 . . . 4  |-  ( x  e.  ( { A }  u.  { B } )  <->  ( x  e.  { A }  \/  x  e.  { B } ) )
3 elsn 3994 . . . . 5  |-  ( x  e.  { A }  <->  x  =  A )
4 elsn 3994 . . . . 5  |-  ( x  e.  { B }  <->  x  =  B )
53, 4orbi12i 528 . . . 4  |-  ( ( x  e.  { A }  \/  x  e.  { B } )  <->  ( x  =  A  \/  x  =  B ) )
62, 5bitri 257 . . 3  |-  ( x  e.  ( { A }  u.  { B } )  <->  ( x  =  A  \/  x  =  B ) )
76abbi2i 2577 . 2  |-  ( { A }  u.  { B } )  =  {
x  |  ( x  =  A  \/  x  =  B ) }
81, 7eqtri 2484 1  |-  { A ,  B }  =  {
x  |  ( x  =  A  \/  x  =  B ) }
Colors of variables: wff setvar class
Syntax hints:    \/ wo 374    = wceq 1455    e. wcel 1898   {cab 2448    u. cun 3414   {csn 3980   {cpr 3982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-v 3059  df-un 3421  df-sn 3981  df-pr 3983
This theorem is referenced by:  elprg  3996  nfpr  4031  pwpw0  4133  pwsn  4206  pwsnALT  4207  zfpair  4651  grothprimlem  9284  nb3graprlem1  25228  nb3grprlem1  39504
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