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Theorem dfpr2 4646
Description: Alternate definition of a 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 4629 . 2 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
2 elun 4153 . . . 4 (𝑥 ∈ ({𝐴} ∪ {𝐵}) ↔ (𝑥 ∈ {𝐴} ∨ 𝑥 ∈ {𝐵}))
3 velsn 4642 . . . . 5 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
4 velsn 4642 . . . . 5 (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
53, 4orbi12i 915 . . . 4 ((𝑥 ∈ {𝐴} ∨ 𝑥 ∈ {𝐵}) ↔ (𝑥 = 𝐴𝑥 = 𝐵))
62, 5bitri 275 . . 3 (𝑥 ∈ ({𝐴} ∪ {𝐵}) ↔ (𝑥 = 𝐴𝑥 = 𝐵))
76eqabi 2877 . 2 ({𝐴} ∪ {𝐵}) = {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵)}
81, 7eqtri 2765 1 {𝐴, 𝐵} = {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵)}
Colors of variables: wff setvar class
Syntax hints:  wo 848   = wceq 1540  wcel 2108  {cab 2714  cun 3949  {csn 4626  {cpr 4628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-un 3956  df-sn 4627  df-pr 4629
This theorem is referenced by:  dfsn2ALT  4647  elprg  4648  nfpr  4692  pwpw0  4813  pwsn  4900  zfpair  5421  grothprimlem  10873  nb3grprlem1  29397  rabsspr  32520  abpr  43422
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