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	$$⊢A\subseteq {\mathbf{C}}_{\mathscr{H}}\to \bigcup A\subseteq {\bigvee }_{\mathscr{H}}\left(A\right)$$

Proof

Step	Hyp	Ref	Expression
1		chsspwh	$$⊢{\mathbf{C}}_{\mathscr{H}}\subseteq 𝒫\mathscr{H}$$
2		sstr	$$⊢\left(A\subseteq {\mathbf{C}}_{\mathscr{H}}\wedge {\mathbf{C}}_{\mathscr{H}}\subseteq 𝒫\mathscr{H}\right)\to A\subseteq 𝒫\mathscr{H}$$
3	1 2	mpan2	$$⊢A\subseteq {\mathbf{C}}_{\mathscr{H}}\to A\subseteq 𝒫\mathscr{H}$$
4		hsupunss	$$⊢A\subseteq 𝒫\mathscr{H}\to \bigcup A\subseteq {\bigvee }_{\mathscr{H}}\left(A\right)$$
5	3 4	syl	$$⊢A\subseteq {\mathbf{C}}_{\mathscr{H}}\to \bigcup A\subseteq {\bigvee }_{\mathscr{H}}\left(A\right)$$