Theorem chsscon1i

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

	Ref	Expression
Hypotheses	ch0le.1	|- A e. CH
	chjcl.2	|- B e. CH
Assertion	chsscon1i	|- ( ( _|_ ` A ) C_ B <-> ( _|_ ` B ) C_ A )

Proof

Step	Hyp	Ref	Expression
1		ch0le.1	|- A e. CH
2		chjcl.2	|- B e. CH
3	1	choccli	|- ( _|_ ` A ) e. CH
4	3 2	chsscon3i	|- ( ( _|_ ` A ) C_ B <-> ( _|_ ` B ) C_ ( _|_ ` ( _|_ ` A ) ) )
5	1	pjococi	|- ( _|_ ` ( _|_ ` A ) ) = A
6	5	sseq2i	|- ( ( _|_ ` B ) C_ ( _|_ ` ( _|_ ` A ) ) <-> ( _|_ ` B ) C_ A )
7	4 6	bitri	|- ( ( _|_ ` A ) C_ B <-> ( _|_ ` B ) C_ A )