Theorem dfpr2 1821

Description: Alternate definition of unordered pair. Definition 5.1 of [TakeutiZaring] p. 15.

Assertion

Ref	Expression
dfpr2	⊢ {A, B} = {x∣(x = A ∨ x = B)}

Distinct variable group(s):   x,A   x,B

Proof of Theorem dfpr2

Step	Hyp	Ref	Expression
1		df-pr 1812	. 2 ⊢ {A, B} = ({A} ∪ {B})
2		elun 1601	. . . 4 ⊢ (x ∈ ({A} ∪ {B}) ↔ (x ∈ {A} ∨ x ∈ {B}))
3		elsn 1820	. . . . 5 ⊢ (x ∈ {A} ↔ x = A)
4		elsn 1820	. . . . 5 ⊢ (x ∈ {B} ↔ x = B)
5	3, 4	orbi12i 216	. . . 4 ⊢ ((x ∈ {A} ∨ x ∈ {B}) ↔ (x = A ∨ x = B))
6	2, 5	bitr 151	. . 3 ⊢ (x ∈ ({A} ∪ {B}) ↔ (x = A ∨ x = B))
7	6	biabri 1180	. 2 ⊢ ({A} ∪ {B}) = {x∣(x = A ∨ x = B)}
8	1, 7	eqtr 1119	1 ⊢ {A, B} = {x∣(x = A ∨ x = B)}

Syntax hints:   ∨ wo 195  {cab 1090   = wceq 1091   ∈ wcel 1092   ∪ cun 1485  {csn 1808  {cpr 1809

This theorem is referenced by:  elprg 1822  pwpw0 1883  zfpair 1891

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-sn 1811  df-pr 1812

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