Theorem shjcli

Description: Closure of CH join. (Contributed by NM, 19-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	shincl.1	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}$$
	shincl.2	$$⊢B\in {\mathbf{S}}_{\mathscr{H}}$$
Assertion	shjcli	$$⊢A{\vee }_{\mathscr{H}}B\in {\mathbf{C}}_{\mathscr{H}}$$

Proof

Step	Hyp	Ref	Expression
1		shincl.1	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}$$
2		shincl.2	$$⊢B\in {\mathbf{S}}_{\mathscr{H}}$$
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	mp2an	$$⊢A{\vee }_{\mathscr{H}}B\in {\mathbf{C}}_{\mathscr{H}}$$