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Metamath Proof Explorer

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 ) )