Theorem qdass

Description: Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016)

	Ref	Expression
Assertion	qdass	$$⊢\left\{A,B\right\}\cup \left\{C,D\right\}=\left\{A,B,C\right\}\cup \left\{D\right\}$$

Proof

Step	Hyp	Ref	Expression
1		unass	$$⊢\left(\left\{A,B\right\}\cup \left\{C\right\}\right)\cup \left\{D\right\}=\left\{A,B\right\}\cup \left(\left\{C\right\}\cup \left\{D\right\}\right)$$
2		df-tp	$$⊢\left\{A,B,C\right\}=\left\{A,B\right\}\cup \left\{C\right\}$$
3	2	uneq1i	$$⊢\left\{A,B,C\right\}\cup \left\{D\right\}=\left(\left\{A,B\right\}\cup \left\{C\right\}\right)\cup \left\{D\right\}$$
4		df-pr	$$⊢\left\{C,D\right\}=\left\{C\right\}\cup \left\{D\right\}$$
5	4	uneq2i	$$⊢\left\{A,B\right\}\cup \left\{C,D\right\}=\left\{A,B\right\}\cup \left(\left\{C\right\}\cup \left\{D\right\}\right)$$
6	1 3 5	3eqtr4ri	$$⊢\left\{A,B\right\}\cup \left\{C,D\right\}=\left\{A,B,C\right\}\cup \left\{D\right\}$$