Theorem chjcl

Description: Closure of join in CH . (Contributed by NM, 2-Nov-1999) (New usage is discouraged.)

	Ref	Expression
Assertion	chjcl	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to A{\vee }_{\mathscr{H}}B\in {\mathbf{C}}_{\mathscr{H}}$$

Proof

Step	Hyp	Ref	Expression
1		chsh	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}\to A\in {\mathbf{S}}_{\mathscr{H}}$$
2		chsh	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}\to B\in {\mathbf{S}}_{\mathscr{H}}$$
3		shjcl	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\right)\to A{\vee }_{\mathscr{H}}B\in {\mathbf{C}}_{\mathscr{H}}$$
4	1 2 3	syl2an	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to A{\vee }_{\mathscr{H}}B\in {\mathbf{C}}_{\mathscr{H}}$$