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Theorem dfpr2 3749
 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 3742 . 2 {A, B} = ({A} ∪ {B})
2 elun 3220 . . . 4 (x ({A} ∪ {B}) ↔ (x {A} x {B}))
3 elsn 3748 . . . . 5 (x {A} ↔ x = A)
4 elsn 3748 . . . . 5 (x {B} ↔ x = B)
53, 4orbi12i 507 . . . 4 ((x {A} x {B}) ↔ (x = A x = B))
62, 5bitri 240 . . 3 (x ({A} ∪ {B}) ↔ (x = A x = B))
76abbi2i 2464 . 2 ({A} ∪ {B}) = {x (x = A x = B)}
81, 7eqtri 2373 1 {A, B} = {x (x = A x = B)}
 Colors of variables: wff setvar class Syntax hints:   ∨ wo 357   = wceq 1642   ∈ wcel 1710  {cab 2339   ∪ cun 3207  {csn 3737  {cpr 3738 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742 This theorem is referenced by:  elprg  3750  nfpr  3773  pwpw0  3855  pwsn  3881  pwsnALT  3882
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