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 = 𝐴 ∨ 𝑥 = 𝐵)}

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem dfpr2

Step	Hyp	Ref	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)
5	3, 4	orbi12i 915	. . . 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 2877	. 2 ⊢ ({𝐴} ∪ {𝐵}) = {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)}
8	1, 7	eqtri 2765	1 ⊢ {𝐴, 𝐵} = {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)}

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