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Theorem dfpr2 4648
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 4632 . 2 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
2 elun 4147 . . . 4 (𝑥 ∈ ({𝐴} ∪ {𝐵}) ↔ (𝑥 ∈ {𝐴} ∨ 𝑥 ∈ {𝐵}))
3 velsn 4645 . . . . 5 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
4 velsn 4645 . . . . 5 (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
53, 4orbi12i 913 . . . 4 ((𝑥 ∈ {𝐴} ∨ 𝑥 ∈ {𝐵}) ↔ (𝑥 = 𝐴𝑥 = 𝐵))
62, 5bitri 275 . . 3 (𝑥 ∈ ({𝐴} ∪ {𝐵}) ↔ (𝑥 = 𝐴𝑥 = 𝐵))
76eqabi 2865 . 2 ({𝐴} ∪ {𝐵}) = {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵)}
81, 7eqtri 2756 1 {𝐴, 𝐵} = {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵)}
Colors of variables: wff setvar class
Syntax hints:  wo 846   = wceq 1534  wcel 2099  {cab 2705  cun 3945  {csn 4629  {cpr 4631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3473  df-un 3952  df-sn 4630  df-pr 4632
This theorem is referenced by:  dfsn2ALT  4649  elprg  4650  nfpr  4695  pwpw0  4817  pwsn  4901  zfpair  5421  grothprimlem  10856  nb3grprlem1  29192  abpr  42839
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