Theorem hsupunss

Description: The union of a set of Hilbert space subsets is smaller than its supremum. (Contributed by NM, 24-Nov-2004) (Revised by Mario Carneiro, 15-May-2014) (New usage is discouraged.)

	Ref	Expression
Assertion	hsupunss	⊢ ( 𝐴 ⊆ 𝒫 ℋ → ∪ 𝐴 ⊆ ( ∨ℋ ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		sspwuni	⊢ ( 𝐴 ⊆ 𝒫 ℋ ↔ ∪ 𝐴 ⊆ ℋ )
2		ococss	⊢ ( ∪ 𝐴 ⊆ ℋ → ∪ 𝐴 ⊆ ( ⊥ ‘ ( ⊥ ‘ ∪ 𝐴 ) ) )
3	1 2	sylbi	⊢ ( 𝐴 ⊆ 𝒫 ℋ → ∪ 𝐴 ⊆ ( ⊥ ‘ ( ⊥ ‘ ∪ 𝐴 ) ) )
4		hsupval	⊢ ( 𝐴 ⊆ 𝒫 ℋ → ( ∨ℋ ‘ 𝐴 ) = ( ⊥ ‘ ( ⊥ ‘ ∪ 𝐴 ) ) )
5	3 4	sseqtrrd	⊢ ( 𝐴 ⊆ 𝒫 ℋ → ∪ 𝐴 ⊆ ( ∨ℋ ‘ 𝐴 ) )