Theorem tpcoma

Description: Swap 1st and 2nd members of an unordered triple. (Contributed by NM, 22-May-2015)

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

Proof

Step	Hyp	Ref	Expression
1		prcom	$$⊢\left\{A,B\right\}=\left\{B,A\right\}$$
2	1	uneq1i	$$⊢\left\{A,B\right\}\cup \left\{C\right\}=\left\{B,A\right\}\cup \left\{C\right\}$$
3		df-tp	$$⊢\left\{A,B,C\right\}=\left\{A,B\right\}\cup \left\{C\right\}$$
4		df-tp	$$⊢\left\{B,A,C\right\}=\left\{B,A\right\}\cup \left\{C\right\}$$
5	2 3 4	3eqtr4i	$$⊢\left\{A,B,C\right\}=\left\{B,A,C\right\}$$