Theorem chsupcl

Description: Closure of supremum of subset of CH . Definition of supremum in Proposition 1 of Kalmbach p. 65. Shows that CH is a complete lattice. Also part of Definition 3.4-1 in MegPav2000 p. 2345 (PDF p. 8). (Contributed by NM, 10-Nov-1999) (New usage is discouraged.)

	Ref	Expression
Assertion	chsupcl	$$⊢A\subseteq {\mathbf{C}}_{\mathscr{H}}\to {\bigvee }_{\mathscr{H}}\left(A\right)\in {\mathbf{C}}_{\mathscr{H}}$$

Proof

Step	Hyp	Ref	Expression
1		chsspwh	$$⊢{\mathbf{C}}_{\mathscr{H}}\subseteq 𝒫\mathscr{H}$$
2		sstr2	$$⊢A\subseteq {\mathbf{C}}_{\mathscr{H}}\to \left({\mathbf{C}}_{\mathscr{H}}\subseteq 𝒫\mathscr{H}\to A\subseteq 𝒫\mathscr{H}\right)$$
3	1 2	mpi	$$⊢A\subseteq {\mathbf{C}}_{\mathscr{H}}\to A\subseteq 𝒫\mathscr{H}$$
4		hsupcl	$$⊢A\subseteq 𝒫\mathscr{H}\to {\bigvee }_{\mathscr{H}}\left(A\right)\in {\mathbf{C}}_{\mathscr{H}}$$
5	3 4	syl	$$⊢A\subseteq {\mathbf{C}}_{\mathscr{H}}\to {\bigvee }_{\mathscr{H}}\left(A\right)\in {\mathbf{C}}_{\mathscr{H}}$$