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Metamath Proof Explorer

Theorem ococss

Description: Inclusion in complement of complement. Part of Proposition 1 of Kalmbach p. 65. (Contributed by NM, 9-Aug-2000) (New usage is discouraged.)

Ref Expression
Assertion ococss 
|- ( A C_ ~H -> A C_ ( _|_ ` ( _|_ ` A ) ) )

Proof

Step Hyp Ref Expression
1 ssel 
 |-  ( A C_ ~H -> ( y e. A -> y e. ~H ) )
2 ocorth 
 |-  ( A C_ ~H -> ( ( y e. A /\ x e. ( _|_ ` A ) ) -> ( y .ih x ) = 0 ) )
3 2 expd 
 |-  ( A C_ ~H -> ( y e. A -> ( x e. ( _|_ ` A ) -> ( y .ih x ) = 0 ) ) )
4 3 ralrimdv 
 |-  ( A C_ ~H -> ( y e. A -> A. x e. ( _|_ ` A ) ( y .ih x ) = 0 ) )
5 1 4 jcad 
 |-  ( A C_ ~H -> ( y e. A -> ( y e. ~H /\ A. x e. ( _|_ ` A ) ( y .ih x ) = 0 ) ) )
6 ocss 
 |-  ( A C_ ~H -> ( _|_ ` A ) C_ ~H )
7 ocel 
 |-  ( ( _|_ ` A ) C_ ~H -> ( y e. ( _|_ ` ( _|_ ` A ) ) <-> ( y e. ~H /\ A. x e. ( _|_ ` A ) ( y .ih x ) = 0 ) ) )
8 6 7 syl 
 |-  ( A C_ ~H -> ( y e. ( _|_ ` ( _|_ ` A ) ) <-> ( y e. ~H /\ A. x e. ( _|_ ` A ) ( y .ih x ) = 0 ) ) )
9 5 8 sylibrd 
 |-  ( A C_ ~H -> ( y e. A -> y e. ( _|_ ` ( _|_ ` A ) ) ) )
10 9 ssrdv 
 |-  ( A C_ ~H -> A C_ ( _|_ ` ( _|_ ` A ) ) )