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Theorem dfpr2 3383
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 3374 . 2 {A, B} = ({A} ∪ {B})
2 elun 3078 . . . 4 (x ({A} ∪ {B}) ↔ (x {A} x {B}))
3 elsn 3382 . . . . 5 (x {A} ↔ x = A)
4 elsn 3382 . . . . 5 (x {B} ↔ x = B)
53, 4orbi12i 680 . . . 4 ((x {A} x {B}) ↔ (x = A x = B))
62, 5bitri 173 . . 3 (x ({A} ∪ {B}) ↔ (x = A x = B))
76abbi2i 2149 . 2 ({A} ∪ {B}) = {x ∣ (x = A x = B)}
81, 7eqtri 2057 1 {A, B} = {x ∣ (x = A x = B)}
Colors of variables: wff set class
Syntax hints:   wo 628   = wceq 1242   wcel 1390  {cab 2023  cun 2909  {csn 3367  {cpr 3368
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374
This theorem is referenced by:  elprg  3384  nfpr  3411  pwsnss  3565
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