Theorem chsupunss

Description: The union of a set of closed subspaces is smaller than its supremum. (Contributed by NM, 14-Aug-2002) (New usage is discouraged.)

	Ref	Expression
Assertion	chsupunss	⊢ ( 𝐴 ⊆ Cℋ → ∪ 𝐴 ⊆ ( ∨ℋ ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		chsspwh	⊢ Cℋ ⊆ 𝒫 ℋ
2		sstr	⊢ ( ( 𝐴 ⊆ Cℋ ∧ Cℋ ⊆ 𝒫 ℋ ) → 𝐴 ⊆ 𝒫 ℋ )
3	1 2	mpan2	⊢ ( 𝐴 ⊆ Cℋ → 𝐴 ⊆ 𝒫 ℋ )
4		hsupunss	⊢ ( 𝐴 ⊆ 𝒫 ℋ → ∪ 𝐴 ⊆ ( ∨ℋ ‘ 𝐴 ) )
5	3 4	syl	⊢ ( 𝐴 ⊆ Cℋ → ∪ 𝐴 ⊆ ( ∨ℋ ‘ 𝐴 ) )