Theorem occon2i

Description: Double contraposition for orthogonal complement. (Contributed by NM, 9-Aug-2000) (New usage is discouraged.)

	Ref	Expression
Hypotheses	occon2.1	$$⊢A\subseteq \mathscr{H}$$
	occon2.2	$$⊢B\subseteq \mathscr{H}$$
Assertion	occon2i	$$⊢A\subseteq B\to \perp \left(\perp \left(A\right)\right)\subseteq \perp \left(\perp \left(B\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		occon2.1	$$⊢A\subseteq \mathscr{H}$$
2		occon2.2	$$⊢B\subseteq \mathscr{H}$$
3		occon2	$$⊢\left(A\subseteq \mathscr{H}\wedge B\subseteq \mathscr{H}\right)\to \left(A\subseteq B\to \perp \left(\perp \left(A\right)\right)\subseteq \perp \left(\perp \left(B\right)\right)\right)$$
4	1 2 3	mp2an	$$⊢A\subseteq B\to \perp \left(\perp \left(A\right)\right)\subseteq \perp \left(\perp \left(B\right)\right)$$