Theorem xpprsng

Description: The Cartesian product of an unordered pair and a singleton. (Contributed by AV, 20-May-2019)

	Ref	Expression
Assertion	xpprsng	$$⊢\left(A\in V\wedge B\in W\wedge C\in U\right)\to \left\{A,B\right\}\times \left\{C\right\}=\left\{⟨A,C⟩,⟨B,C⟩\right\}$$

Proof

Step	Hyp	Ref	Expression
1		df-pr	$$⊢\left\{A,B\right\}=\left\{A\right\}\cup \left\{B\right\}$$
2	1	xpeq1i	$$⊢\left\{A,B\right\}\times \left\{C\right\}=\left(\left\{A\right\}\cup \left\{B\right\}\right)\times \left\{C\right\}$$
3		xpsng	$$⊢\left(A\in V\wedge C\in U\right)\to \left\{A\right\}\times \left\{C\right\}=\left\{⟨A,C⟩\right\}$$
4	3	3adant2	$$⊢\left(A\in V\wedge B\in W\wedge C\in U\right)\to \left\{A\right\}\times \left\{C\right\}=\left\{⟨A,C⟩\right\}$$
5		xpsng	$$⊢\left(B\in W\wedge C\in U\right)\to \left\{B\right\}\times \left\{C\right\}=\left\{⟨B,C⟩\right\}$$
6	5	3adant1	$$⊢\left(A\in V\wedge B\in W\wedge C\in U\right)\to \left\{B\right\}\times \left\{C\right\}=\left\{⟨B,C⟩\right\}$$
7	4 6	uneq12d	$$⊢\left(A\in V\wedge B\in W\wedge C\in U\right)\to \left(\left\{A\right\}\times \left\{C\right\}\right)\cup \left(\left\{B\right\}\times \left\{C\right\}\right)=\left\{⟨A,C⟩\right\}\cup \left\{⟨B,C⟩\right\}$$
8		xpundir	$$⊢\left(\left\{A\right\}\cup \left\{B\right\}\right)\times \left\{C\right\}=\left(\left\{A\right\}\times \left\{C\right\}\right)\cup \left(\left\{B\right\}\times \left\{C\right\}\right)$$
9		df-pr	$$⊢\left\{⟨A,C⟩,⟨B,C⟩\right\}=\left\{⟨A,C⟩\right\}\cup \left\{⟨B,C⟩\right\}$$
10	7 8 9	3eqtr4g	$$⊢\left(A\in V\wedge B\in W\wedge C\in U\right)\to \left(\left\{A\right\}\cup \left\{B\right\}\right)\times \left\{C\right\}=\left\{⟨A,C⟩,⟨B,C⟩\right\}$$
11	2 10	eqtrid	$$⊢\left(A\in V\wedge B\in W\wedge C\in U\right)\to \left\{A,B\right\}\times \left\{C\right\}=\left\{⟨A,C⟩,⟨B,C⟩\right\}$$