# 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 ${⊢}\left\{{A},{B}\right\}=\left\{{A}\right\}\cup \left\{{B}\right\}$

### Detailed syntax breakdown

Step Hyp Ref Expression
0 cA ${class}{A}$
1 cB ${class}{B}$
2 0 1 cpr ${class}\left\{{A},{B}\right\}$
3 0 csn ${class}\left\{{A}\right\}$
4 1 csn ${class}\left\{{B}\right\}$
5 3 4 cun ${class}\left(\left\{{A}\right\}\cup \left\{{B}\right\}\right)$
6 2 5 wceq ${wff}\left\{{A},{B}\right\}=\left\{{A}\right\}\cup \left\{{B}\right\}$