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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	$⊢ A \subseteq ℋ \to A \subseteq ⊥ (⊥ (A))$

Proof

Step	Hyp	Ref	Expression
1		ssel	$⊢ A \subseteq ℋ \to (y \in A \to y \in ℋ)$
2		ocorth	$⊢ A \subseteq ℋ \to ((y \in A \land x \in ⊥ (A)) \to y \cdot_{ih} x = 0)$
3	2	expd	$⊢ A \subseteq ℋ \to (y \in A \to (x \in ⊥ (A) \to y \cdot_{ih} x = 0))$
4	3	ralrimdv	$⊢ A \subseteq ℋ \to (y \in A \to \forall x \in ⊥ (A) y \cdot_{ih} x = 0)$
5	1 4	jcad	$⊢ A \subseteq ℋ \to (y \in A \to (y \in ℋ \land \forall x \in ⊥ (A) y \cdot_{ih} x = 0))$
6		ocss	$⊢ A \subseteq ℋ \to ⊥ (A) \subseteq ℋ$
7		ocel	$⊢ ⊥ (A) \subseteq ℋ \to (y \in ⊥ (⊥ (A)) \leftrightarrow (y \in ℋ \land \forall x \in ⊥ (A) y \cdot_{ih} x = 0))$
8	6 7	syl	$⊢ A \subseteq ℋ \to (y \in ⊥ (⊥ (A)) \leftrightarrow (y \in ℋ \land \forall x \in ⊥ (A) y \cdot_{ih} x = 0))$
9	5 8	sylibrd	$⊢ A \subseteq ℋ \to (y \in A \to y \in ⊥ (⊥ (A)))$
10	9	ssrdv	$⊢ A \subseteq ℋ \to A \subseteq ⊥ (⊥ (A))$