MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfpr2 Structured version   Visualization version   GIF version

Theorem dfpr2 4650
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 4633 . 2 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
2 elun 4162 . . . 4 (𝑥 ∈ ({𝐴} ∪ {𝐵}) ↔ (𝑥 ∈ {𝐴} ∨ 𝑥 ∈ {𝐵}))
3 velsn 4646 . . . . 5 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
4 velsn 4646 . . . . 5 (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
53, 4orbi12i 914 . . . 4 ((𝑥 ∈ {𝐴} ∨ 𝑥 ∈ {𝐵}) ↔ (𝑥 = 𝐴𝑥 = 𝐵))
62, 5bitri 275 . . 3 (𝑥 ∈ ({𝐴} ∪ {𝐵}) ↔ (𝑥 = 𝐴𝑥 = 𝐵))
76eqabi 2874 . 2 ({𝐴} ∪ {𝐵}) = {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵)}
81, 7eqtri 2762 1 {𝐴, 𝐵} = {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵)}
Colors of variables: wff setvar class
Syntax hints:  wo 847   = wceq 1536  wcel 2105  {cab 2711  cun 3960  {csn 4630  {cpr 4632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-v 3479  df-un 3967  df-sn 4631  df-pr 4633
This theorem is referenced by:  dfsn2ALT  4651  elprg  4652  nfpr  4696  pwpw0  4817  pwsn  4904  zfpair  5426  grothprimlem  10870  nb3grprlem1  29411  rabsspr  32528  abpr  43398
  Copyright terms: Public domain W3C validator