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Metamath Proof Explorer

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 ) ) )