Theorem shococss

Description: Inclusion in complement of complement. Part of Proposition 1 of Kalmbach p. 65. (Contributed by NM, 10-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Assertion	shococss	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}\to A\subseteq \perp \left(\perp \left(A\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		shss	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}\to A\subseteq \mathscr{H}$$
2		ococss	$$⊢A\subseteq \mathscr{H}\to A\subseteq \perp \left(\perp \left(A\right)\right)$$
3	1 2	syl	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}\to A\subseteq \perp \left(\perp \left(A\right)\right)$$