Theorem tpidm12

Description: Unordered triple { A , A , B } is just an overlong way to write { A , B } . (Contributed by David A. Wheeler, 10-May-2015)

	Ref	Expression
Assertion	tpidm12	$$⊢\left\{A,A,B\right\}=\left\{A,B\right\}$$

Proof

Step	Hyp	Ref	Expression
1		dfsn2	$$⊢\left\{A\right\}=\left\{A,A\right\}$$
2	1	uneq1i	$$⊢\left\{A\right\}\cup \left\{B\right\}=\left\{A,A\right\}\cup \left\{B\right\}$$
3		df-pr	$$⊢\left\{A,B\right\}=\left\{A\right\}\cup \left\{B\right\}$$
4		df-tp	$$⊢\left\{A,A,B\right\}=\left\{A,A\right\}\cup \left\{B\right\}$$
5	2 3 4	3eqtr4ri	$$⊢\left\{A,A,B\right\}=\left\{A,B\right\}$$