Theorem mress

Description: A Moore-closed subset is a subset. (Contributed by Stefan O'Rear, 31-Jan-2015)

	Ref	Expression
Assertion	mress	⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑆 ∈ 𝐶 ) → 𝑆 ⊆ 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		mresspw	⊢ ( 𝐶 ∈ ( Moore ‘ 𝑋 ) → 𝐶 ⊆ 𝒫 𝑋 )
2	1	sselda	⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑆 ∈ 𝐶 ) → 𝑆 ∈ 𝒫 𝑋 )
3	2	elpwid	⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑆 ∈ 𝐶 ) → 𝑆 ⊆ 𝑋 )