Theorem chsscon2i

Description: Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ch0le.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
	chjcl.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	chsscon2i	$$⊢A\subseteq \perp \left(B\right)\leftrightarrow B\subseteq \perp \left(A\right)$$

Proof

Step	Hyp	Ref	Expression
1		ch0le.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
2		chjcl.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
3	1	chssii	$$⊢A\subseteq \mathscr{H}$$
4	2	chssii	$$⊢B\subseteq \mathscr{H}$$
5		occon3	$$⊢\left(A\subseteq \mathscr{H}\wedge B\subseteq \mathscr{H}\right)\to \left(A\subseteq \perp \left(B\right)\leftrightarrow B\subseteq \perp \left(A\right)\right)$$
6	3 4 5	mp2an	$$⊢A\subseteq \perp \left(B\right)\leftrightarrow B\subseteq \perp \left(A\right)$$