Theorem chpsscon3

Description: Hilbert lattice contraposition law for strict ordering. (Contributed by NM, 12-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	chpsscon3	|- ( ( A e. CH /\ B e. CH ) -> ( A C. B <-> ( _|_ ` B ) C. ( _|_ ` A ) ) )

Proof

Step	Hyp	Ref	Expression
1		chsscon3	|- ( ( A e. CH /\ B e. CH ) -> ( A C_ B <-> ( _|_ ` B ) C_ ( _|_ ` A ) ) )
2		chsscon3	|- ( ( B e. CH /\ A e. CH ) -> ( B C_ A <-> ( _|_ ` A ) C_ ( _|_ ` B ) ) )
3	2	ancoms	|- ( ( A e. CH /\ B e. CH ) -> ( B C_ A <-> ( _|_ ` A ) C_ ( _|_ ` B ) ) )
4	3	notbid	|- ( ( A e. CH /\ B e. CH ) -> ( -. B C_ A <-> -. ( _|_ ` A ) C_ ( _|_ ` B ) ) )
5	1 4	anbi12d	|- ( ( A e. CH /\ B e. CH ) -> ( ( A C_ B /\ -. B C_ A ) <-> ( ( _|_ ` B ) C_ ( _|_ ` A ) /\ -. ( _|_ ` A ) C_ ( _|_ ` B ) ) ) )
6		dfpss3	|- ( A C. B <-> ( A C_ B /\ -. B C_ A ) )
7		dfpss3	|- ( ( _|_ ` B ) C. ( _|_ ` A ) <-> ( ( _|_ ` B ) C_ ( _|_ ` A ) /\ -. ( _|_ ` A ) C_ ( _|_ ` B ) ) )
8	5 6 7	3bitr4g	|- ( ( A e. CH /\ B e. CH ) -> ( A C. B <-> ( _|_ ` B ) C. ( _|_ ` A ) ) )