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Metamath Proof Explorer

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 ⊢ φ → A ∈ Moore ⁡ X
mrcssd.2 ⊢ N = mrCls ⁡ A
Assertion mrcssvd ⊢ φ → N ⁡ B ⊆ X

Proof

Step Hyp Ref Expression
1 mrcssd.1 ⊢ φ → A ∈ Moore ⁡ X
2 mrcssd.2 ⊢ N = mrCls ⁡ A
3 2 mrcssv ⊢ A ∈ Moore ⁡ X → N ⁡ B ⊆ X
4 1 3 syl ⊢ φ → N ⁡ B ⊆ X