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Metamath Proof Explorer

Theorem chpsscon2

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

Ref Expression
Assertion chpsscon2 
|- ( ( A e. CH /\ B e. CH ) -> ( A C. ( _|_ ` B ) <-> B C. ( _|_ ` A ) ) )

Proof

Step Hyp Ref Expression
1 choccl 
 |-  ( B e. CH -> ( _|_ ` B ) e. CH )
2 chpsscon3 
 |-  ( ( A e. CH /\ ( _|_ ` B ) e. CH ) -> ( A C. ( _|_ ` B ) <-> ( _|_ ` ( _|_ ` B ) ) C. ( _|_ ` A ) ) )
3 1 2 sylan2 
 |-  ( ( A e. CH /\ B e. CH ) -> ( A C. ( _|_ ` B ) <-> ( _|_ ` ( _|_ ` B ) ) C. ( _|_ ` A ) ) )
4 ococ 
 |-  ( B e. CH -> ( _|_ ` ( _|_ ` B ) ) = B )
5 4 adantl 
 |-  ( ( A e. CH /\ B e. CH ) -> ( _|_ ` ( _|_ ` B ) ) = B )
6 5 psseq1d 
 |-  ( ( 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 ) ) )