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 ) ) |
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 ) ) |