Theorem shjval

Description: Value of join in SH . (Contributed by NM, 9-Aug-2000) (New usage is discouraged.)

	Ref	Expression
Assertion	shjval	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\right)\to A{\vee }_{\mathscr{H}}B=\perp \left(\perp \left(A\cup B\right)\right)$$

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		sshjval	$$⊢\left(A\subseteq \mathscr{H}\wedge B\subseteq \mathscr{H}\right)\to A{\vee }_{\mathscr{H}}B=\perp \left(\perp \left(A\cup B\right)\right)$$
4	1 2 3	syl2an	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\right)\to A{\vee }_{\mathscr{H}}B=\perp \left(\perp \left(A\cup B\right)\right)$$