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Theorem dfpr2 4355
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 4339 . 2 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
2 elun 3917 . . . 4 (𝑥 ∈ ({𝐴} ∪ {𝐵}) ↔ (𝑥 ∈ {𝐴} ∨ 𝑥 ∈ {𝐵}))
3 velsn 4352 . . . . 5 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
4 velsn 4352 . . . . 5 (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
53, 4orbi12i 938 . . . 4 ((𝑥 ∈ {𝐴} ∨ 𝑥 ∈ {𝐵}) ↔ (𝑥 = 𝐴𝑥 = 𝐵))
62, 5bitri 266 . . 3 (𝑥 ∈ ({𝐴} ∪ {𝐵}) ↔ (𝑥 = 𝐴𝑥 = 𝐵))
76abbi2i 2881 . 2 ({𝐴} ∪ {𝐵}) = {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵)}
81, 7eqtri 2787 1 {𝐴, 𝐵} = {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵)}
Colors of variables: wff setvar class
Syntax hints:  wo 873   = wceq 1652  wcel 2155  {cab 2751  cun 3732  {csn 4336  {cpr 4338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-un 3739  df-sn 4337  df-pr 4339
This theorem is referenced by:  dfsn2ALT  4356  elprg  4357  nfpr  4390  pwpw0  4500  pwsn  4588  pwsnALT  4589  zfpair  5062  grothprimlem  9910  nb3grprlem1  26564
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