# Metamath Proof Explorer

## Definition df-pr

Description: Define unordered pair of classes. Definition 7.1 of Quine p. 48. For example, A e. { 1 , -u 1 } -> ( A ^ 2 ) = 1 ( ex-pr ). They are unordered, so { A , B } = { B , A } as proven by prcom . For a more traditional definition, but requiring a dummy variable, see dfpr2 . { A , A } is also an unordered pair, but also a singleton because of { A } = { A , A } (see dfsn2 ). Therefore, { A , B } is called aproper (unordered) pair iff A =/= B and A and B are sets.

Note: ordered pairs are a completely different object defined below in df-op . When the term "pair" is used without qualifier, it generally means "unordered pair", and the context makes it clear which version is meant. (Contributed by NM, 21-Jun-1993)

Ref Expression
Assertion df-pr { 𝐴 , 𝐵 } = ( { 𝐴 } ∪ { 𝐵 } )

### Detailed syntax breakdown

Step Hyp Ref Expression
0 cA 𝐴
1 cB 𝐵
2 0 1 cpr { 𝐴 , 𝐵 }
3 0 csn { 𝐴 }
4 1 csn { 𝐵 }
5 3 4 cun ( { 𝐴 } ∪ { 𝐵 } )
6 2 5 wceq { 𝐴 , 𝐵 } = ( { 𝐴 } ∪ { 𝐵 } )