Theorem prelpw

Description: An unordered pair of two sets is a member of the powerclass of a class if and only if the two sets are members of that class. (Contributed by AV, 8-Jan-2020)

	Ref	Expression
Assertion	prelpw	$$⊢\left(A\in V\wedge B\in W\right)\to \left(\left(A\in C\wedge B\in C\right)\leftrightarrow \left\{A,B\right\}\in 𝒫C\right)$$

Proof

Step	Hyp	Ref	Expression
1		prssg	$$⊢\left(A\in V\wedge B\in W\right)\to \left(\left(A\in C\wedge B\in C\right)\leftrightarrow \left\{A,B\right\}\subseteq C\right)$$
2		prex	$$⊢\left\{A,B\right\}\in \mathrm{V}$$
3	2	elpw	$$⊢\left\{A,B\right\}\in 𝒫C\leftrightarrow \left\{A,B\right\}\subseteq C$$
4	1 3	bitr4di	$$⊢\left(A\in V\wedge B\in W\right)\to \left(\left(A\in C\wedge B\in C\right)\leftrightarrow \left\{A,B\right\}\in 𝒫C\right)$$