Theorem dfpr2 3749

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 3742	. 2 ⊢ {A, B} = ({A} ∪ {B})
2		elun 3220	. . . 4 ⊢ (x ∈ ({A} ∪ {B}) ↔ (x ∈ {A} ∨ x ∈ {B}))
3		elsn 3748	. . . . 5 ⊢ (x ∈ {A} ↔ x = A)
4		elsn 3748	. . . . 5 ⊢ (x ∈ {B} ↔ x = B)
5	3, 4	orbi12i 507	. . . 4 ⊢ ((x ∈ {A} ∨ x ∈ {B}) ↔ (x = A ∨ x = B))
6	2, 5	bitri 240	. . 3 ⊢ (x ∈ ({A} ∪ {B}) ↔ (x = A ∨ x = B))
7	6	abbi2i 2464	. 2 ⊢ ({A} ∪ {B}) = {x ∣ (x = A ∨ x = B)}
8	1, 7	eqtri 2373	1 ⊢ {A, B} = {x ∣ (x = A ∨ x = B)}

Syntax hints:   ∨ wo 357   = wceq 1642   ∈ wcel 1710  {cab 2339   ∪ cun 3207  {csn 3737  {cpr 3738

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742

This theorem is referenced by:  elprg  3750  nfpr  3773  pwpw0  3855  pwsn  3881  pwsnALT  3882