Theorem mrcssvd

Description: The Moore closure of a set is a subset of the base. Deduction form of mrcssv . (Contributed by David Moews, 1-May-2017)

	Ref	Expression
Hypotheses	mrcssd.1	$$⊢\phi \to A\in \mathrm{Moore}\left(X\right)$$
	mrcssd.2	$$⊢N=\mathrm{mrCls}\left(A\right)$$
Assertion	mrcssvd	$$⊢\phi \to N\left(B\right)\subseteq X$$

Proof

Step	Hyp	Ref	Expression
1		mrcssd.1	$$⊢\phi \to A\in \mathrm{Moore}\left(X\right)$$
2		mrcssd.2	$$⊢N=\mathrm{mrCls}\left(A\right)$$
3	2	mrcssv	$$⊢A\in \mathrm{Moore}\left(X\right)\to N\left(B\right)\subseteq X$$
4	1 3	syl	$$⊢\phi \to N\left(B\right)\subseteq X$$