Theorem mresspw

Description: A Moore collection is a subset of the power of the base set; each closed subset of the system is actually a subset of the base. (Contributed by Stefan O'Rear, 30-Jan-2015)

	Ref	Expression
Assertion	mresspw	$$⊢C\in \mathrm{Moore}\left(X\right)\to C\subseteq 𝒫X$$

Proof

Step	Hyp	Ref	Expression
1		ismre	$$⊢C\in \mathrm{Moore}\left(X\right)\leftrightarrow \left(C\subseteq 𝒫X\wedge X\in C\wedge \forall s\in 𝒫C\left(s\ne \varnothing \to \bigcap s\in C\right)\right)$$
2	1	simp1bi	$$⊢C\in \mathrm{Moore}\left(X\right)\to C\subseteq 𝒫X$$