Theorem chsscon1

Description: Hilbert lattice contraposition law. (Contributed by NM, 21-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	chsscon1	|- ( ( A e. CH /\ B e. CH ) -> ( ( _|_ ` A ) C_ B <-> ( _|_ ` B ) C_ A ) )

Proof

Step	Hyp	Ref	Expression
1		choccl	|- ( A e. CH -> ( _|_ ` A ) e. CH )
2		chsscon3	|- ( ( ( _|_ ` A ) e. CH /\ B e. CH ) -> ( ( _|_ ` A ) C_ B <-> ( _|_ ` B ) C_ ( _|_ ` ( _|_ ` A ) ) ) )
3	1 2	sylan	|- ( ( A e. CH /\ B e. CH ) -> ( ( _|_ ` A ) C_ B <-> ( _|_ ` B ) C_ ( _|_ ` ( _|_ ` A ) ) ) )
4		ococ	|- ( A e. CH -> ( _|_ ` ( _|_ ` A ) ) = A )
5	4	adantr	|- ( ( A e. CH /\ B e. CH ) -> ( _|_ ` ( _|_ ` A ) ) = A )
6	5	sseq2d	|- ( ( A e. CH /\ B e. CH ) -> ( ( _|_ ` B ) C_ ( _|_ ` ( _|_ ` A ) ) <-> ( _|_ ` B ) C_ A ) )
7	3 6	bitrd	|- ( ( A e. CH /\ B e. CH ) -> ( ( _|_ ` A ) C_ B <-> ( _|_ ` B ) C_ A ) )