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Theorem dfpr2 3512
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 = 𝐴𝑥 = 𝐵)}
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem dfpr2
StepHypRef Expression
1 df-pr 3500 . 2 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
2 elun 3183 . . . 4 (𝑥 ∈ ({𝐴} ∪ {𝐵}) ↔ (𝑥 ∈ {𝐴} ∨ 𝑥 ∈ {𝐵}))
3 velsn 3510 . . . . 5 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
4 velsn 3510 . . . . 5 (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
53, 4orbi12i 736 . . . 4 ((𝑥 ∈ {𝐴} ∨ 𝑥 ∈ {𝐵}) ↔ (𝑥 = 𝐴𝑥 = 𝐵))
62, 5bitri 183 . . 3 (𝑥 ∈ ({𝐴} ∪ {𝐵}) ↔ (𝑥 = 𝐴𝑥 = 𝐵))
76abbi2i 2229 . 2 ({𝐴} ∪ {𝐵}) = {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵)}
81, 7eqtri 2135 1 {𝐴, 𝐵} = {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵)}
Colors of variables: wff set class
Syntax hints:  wo 680   = wceq 1314  wcel 1463  {cab 2101  cun 3035  {csn 3493  {cpr 3494
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-un 3041  df-sn 3499  df-pr 3500
This theorem is referenced by:  elprg  3513  nfpr  3539  pwsnss  3696  minmax  10893  xrminmax  10926
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