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Metamath Proof Explorer

Theorem ssdmd2

Description: Ordering implies the dual modular pair property. Remark in MaedaMaeda p. 1. (Contributed by NM, 22-Jun-2004) (New usage is discouraged.)

Ref Expression
Assertion ssdmd2 
|- ( ( A e. CH /\ B e. CH /\ A C_ B ) -> ( _|_ ` B ) MH ( _|_ ` A ) )

Proof

Step Hyp Ref Expression
1 chsscon3 
 |-  ( ( A e. CH /\ B e. CH ) -> ( A C_ B <-> ( _|_ ` B ) C_ ( _|_ ` A ) ) )
2 choccl 
 |-  ( B e. CH -> ( _|_ ` B ) e. CH )
3 choccl 
 |-  ( A e. CH -> ( _|_ ` A ) e. CH )
4 ssmd1 
 |-  ( ( ( _|_ ` B ) e. CH /\ ( _|_ ` A ) e. CH /\ ( _|_ ` B ) C_ ( _|_ ` A ) ) -> ( _|_ ` B ) MH ( _|_ ` A ) )
5 4 3expia 
 |-  ( ( ( _|_ ` B ) e. CH /\ ( _|_ ` A ) e. CH ) -> ( ( _|_ ` B ) C_ ( _|_ ` A ) -> ( _|_ ` B ) MH ( _|_ ` A ) ) )
6 2 3 5 syl2anr 
 |-  ( ( A e. CH /\ B e. CH ) -> ( ( _|_ ` B ) C_ ( _|_ ` A ) -> ( _|_ ` B ) MH ( _|_ ` A ) ) )
7 1 6 sylbid 
 |-  ( ( A e. CH /\ B e. CH ) -> ( A C_ B -> ( _|_ ` B ) MH ( _|_ ` A ) ) )
8 7 3impia 
 |-  ( ( A e. CH /\ B e. CH /\ A C_ B ) -> ( _|_ ` B ) MH ( _|_ ` A ) )