Theorem prsspw

Description: An unordered pair belongs to the power class of a class iff each member belongs to the class. (Contributed by NM, 10-Dec-2003) (Proof shortened by Andrew Salmon, 26-Jun-2011) (Proof shortened by OpenAI, 25-Mar-2020)

	Ref	Expression
Hypotheses	prsspw.1	$$⊢A\in \mathrm{V}$$
	prsspw.2	$$⊢B\in \mathrm{V}$$
Assertion	prsspw	$$⊢\left\{A,B\right\}\subseteq 𝒫C\leftrightarrow \left(A\subseteq C\wedge B\subseteq C\right)$$

Proof

Step	Hyp	Ref	Expression
1		prsspw.1	$$⊢A\in \mathrm{V}$$
2		prsspw.2	$$⊢B\in \mathrm{V}$$
3		prsspwg	$$⊢\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)\to \left(\left\{A,B\right\}\subseteq 𝒫C\leftrightarrow \left(A\subseteq C\wedge B\subseteq C\right)\right)$$
4	1 2 3	mp2an	$$⊢\left\{A,B\right\}\subseteq 𝒫C\leftrightarrow \left(A\subseteq C\wedge B\subseteq C\right)$$