Theorem qdassr

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

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

Proof

Step	Hyp	Ref	Expression
1		unass	$$⊢\left(\left\{A\right\}\cup \left\{B\right\}\right)\cup \left\{C,D\right\}=\left\{A\right\}\cup \left(\left\{B\right\}\cup \left\{C,D\right\}\right)$$
2		df-pr	$$⊢\left\{A,B\right\}=\left\{A\right\}\cup \left\{B\right\}$$
3	2	uneq1i	$$⊢\left\{A,B\right\}\cup \left\{C,D\right\}=\left(\left\{A\right\}\cup \left\{B\right\}\right)\cup \left\{C,D\right\}$$
4		tpass	$$⊢\left\{B,C,D\right\}=\left\{B\right\}\cup \left\{C,D\right\}$$
5	4	uneq2i	$$⊢\left\{A\right\}\cup \left\{B,C,D\right\}=\left\{A\right\}\cup \left(\left\{B\right\}\cup \left\{C,D\right\}\right)$$
6	1 3 5	3eqtr4i	$$⊢\left\{A,B\right\}\cup \left\{C,D\right\}=\left\{A\right\}\cup \left\{B,C,D\right\}$$