Theorem shjcl

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

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

Proof

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