Theorem prcom

Description: Commutative law for unordered pairs. (Contributed by NM, 15-Jul-1993)

	Ref	Expression
Assertion	prcom	$$⊢\left\{A,B\right\}=\left\{B,A\right\}$$

Proof

Step	Hyp	Ref	Expression
1		uncom	$$⊢\left\{A\right\}\cup \left\{B\right\}=\left\{B\right\}\cup \left\{A\right\}$$
2		df-pr	$$⊢\left\{A,B\right\}=\left\{A\right\}\cup \left\{B\right\}$$
3		df-pr	$$⊢\left\{B,A\right\}=\left\{B\right\}\cup \left\{A\right\}$$
4	1 2 3	3eqtr4i	$$⊢\left\{A,B\right\}=\left\{B,A\right\}$$