Theorem chsscon1i

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	chsscon1i	$$⊢\perp \left(A\right)\subseteq B\leftrightarrow \perp \left(B\right)\subseteq A$$

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	choccli	$$⊢\perp \left(A\right)\in {\mathbf{C}}_{\mathscr{H}}$$
4	3 2	chsscon3i	$$⊢\perp \left(A\right)\subseteq B\leftrightarrow \perp \left(B\right)\subseteq \perp \left(\perp \left(A\right)\right)$$
5	1	pjococi	$$⊢\perp \left(\perp \left(A\right)\right)=A$$
6	5	sseq2i	$$⊢\perp \left(B\right)\subseteq \perp \left(\perp \left(A\right)\right)\leftrightarrow \perp \left(B\right)\subseteq A$$
7	4 6	bitri	$$⊢\perp \left(A\right)\subseteq B\leftrightarrow \perp \left(B\right)\subseteq A$$