Theorem dfpr2 4601

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 = 𝐴 ∨ 𝑥 = 𝐵)}

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem dfpr2

Step	Hyp	Ref	Expression
1		df-pr 4583	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
2		elun 4105	. . . 4 ⊢ (𝑥 ∈ ({𝐴} ∪ {𝐵}) ↔ (𝑥 ∈ {𝐴} ∨ 𝑥 ∈ {𝐵}))
3		velsn 4596	. . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
4		velsn 4596	. . . . 5 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
5	3, 4	orbi12i 914	. . . 4 ⊢ ((𝑥 ∈ {𝐴} ∨ 𝑥 ∈ {𝐵}) ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))
6	2, 5	bitri 275	. . 3 ⊢ (𝑥 ∈ ({𝐴} ∪ {𝐵}) ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))
7	6	eqabi 2871	. 2 ⊢ ({𝐴} ∪ {𝐵}) = {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)}
8	1, 7	eqtri 2759	1 ⊢ {𝐴, 𝐵} = {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)}

Syntax hints:   ∨ wo 847   = wceq 1541   ∈ wcel 2113  {cab 2714   ∪ cun 3899  {csn 4580  {cpr 4582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3442  df-un 3906  df-sn 4581  df-pr 4583

This theorem is referenced by:  dfsn2ALT  4602  elprg  4603  nfpr  4649  pwpw0  4769  pwsn  4856  zfpair  5366  grothprimlem  10744  nb3grprlem1  29453  rabsspr  32576  abpr  43646