Theorem mre1cl

Description: In any Moore collection the base set is closed. (Contributed by Stefan O'Rear, 30-Jan-2015)

	Ref	Expression
Assertion	mre1cl	$$⊢C\in \mathrm{Moore}\left(X\right)\to X\in C$$

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	simp2bi	$$⊢C\in \mathrm{Moore}\left(X\right)\to X\in C$$