MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfpr2 Structured version   Unicode version

Theorem dfpr2 4042
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 4030 . 2  |-  { A ,  B }  =  ( { A }  u.  { B } )
2 elun 3645 . . . 4  |-  ( x  e.  ( { A }  u.  { B } )  <->  ( x  e.  { A }  \/  x  e.  { B } ) )
3 elsn 4041 . . . . 5  |-  ( x  e.  { A }  <->  x  =  A )
4 elsn 4041 . . . . 5  |-  ( x  e.  { B }  <->  x  =  B )
53, 4orbi12i 521 . . . 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 2600 . 2  |-  ( { A }  u.  { B } )  =  {
x  |  ( x  =  A  \/  x  =  B ) }
81, 7eqtri 2496 1  |-  { A ,  B }  =  {
x  |  ( x  =  A  \/  x  =  B ) }
Colors of variables: wff setvar class
Syntax hints:    \/ wo 368    = wceq 1379    e. wcel 1767   {cab 2452    u. cun 3474   {csn 4027   {cpr 4029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-un 3481  df-sn 4028  df-pr 4030
This theorem is referenced by:  elprg  4043  nfpr  4074  pwpw0  4175  pwsn  4239  pwsnALT  4240  zfpair  4684  grothprimlem  9207  nb3graprlem1  24127
  Copyright terms: Public domain W3C validator