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 ∈ ( Moore ‘ 𝑋 ) → 𝑋 ∈ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		ismre	⊢ ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ↔ ( 𝐶 ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ 𝐶 ∧ ∀ 𝑠 ∈ 𝒫 𝐶 ( 𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝐶 ) ) )
2	1	simp2bi	⊢ ( 𝐶 ∈ ( Moore ‘ 𝑋 ) → 𝑋 ∈ 𝐶 )