This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) )