Theorem ssprss

Description: A pair as subset of a pair. (Contributed by AV, 26-Oct-2020)

	Ref	Expression
Assertion	ssprss	$$⊢\left(A\in V\wedge B\in W\right)\to \left(\left\{A,B\right\}\subseteq \left\{C,D\right\}\leftrightarrow \left(\left(A=C\vee A=D\right)\wedge \left(B=C\vee B=D\right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		prssg	$$⊢\left(A\in V\wedge B\in W\right)\to \left(\left(A\in \left\{C,D\right\}\wedge B\in \left\{C,D\right\}\right)\leftrightarrow \left\{A,B\right\}\subseteq \left\{C,D\right\}\right)$$
2		elprg	$$⊢A\in V\to \left(A\in \left\{C,D\right\}\leftrightarrow \left(A=C\vee A=D\right)\right)$$
3		elprg	$$⊢B\in W\to \left(B\in \left\{C,D\right\}\leftrightarrow \left(B=C\vee B=D\right)\right)$$
4	2 3	bi2anan9	$$⊢\left(A\in V\wedge B\in W\right)\to \left(\left(A\in \left\{C,D\right\}\wedge B\in \left\{C,D\right\}\right)\leftrightarrow \left(\left(A=C\vee A=D\right)\wedge \left(B=C\vee B=D\right)\right)\right)$$
5	1 4	bitr3d	$$⊢\left(A\in V\wedge B\in W\right)\to \left(\left\{A,B\right\}\subseteq \left\{C,D\right\}\leftrightarrow \left(\left(A=C\vee A=D\right)\wedge \left(B=C\vee B=D\right)\right)\right)$$