# 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 ) ) )`