Theorem dfpr2 3546

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 = 𝐴 ∨ 𝑥 = 𝐵)}

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem dfpr2

Step	Hyp	Ref	Expression
1		df-pr 3534	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
2		elun 3217	. . . 4 ⊢ (𝑥 ∈ ({𝐴} ∪ {𝐵}) ↔ (𝑥 ∈ {𝐴} ∨ 𝑥 ∈ {𝐵}))
3		velsn 3544	. . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
4		velsn 3544	. . . . 5 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
5	3, 4	orbi12i 753	. . . 4 ⊢ ((𝑥 ∈ {𝐴} ∨ 𝑥 ∈ {𝐵}) ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))
6	2, 5	bitri 183	. . 3 ⊢ (𝑥 ∈ ({𝐴} ∪ {𝐵}) ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))
7	6	abbi2i 2254	. 2 ⊢ ({𝐴} ∪ {𝐵}) = {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)}
8	1, 7	eqtri 2160	1 ⊢ {𝐴, 𝐵} = {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)}

Syntax hints:   ∨ wo 697   = wceq 1331   ∈ wcel 1480  {cab 2125   ∪ cun 3069  {csn 3527  {cpr 3528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534

This theorem is referenced by:  elprg  3547  nfpr  3573  pwsnss  3730  minmax  11001  xrminmax  11034