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Metamath Proof Explorer

Theorem shococss

Description: Inclusion in complement of complement. Part of Proposition 1 of Kalmbach p. 65. (Contributed by NM, 10-Oct-1999) (New usage is discouraged.)

		Ref	Expression
	Assertion	shococss	$⊢ A \in S_{ℋ} \to A \subseteq ⊥ (⊥ (A))$

Proof

Step	Hyp	Ref	Expression
1		shss	$⊢ A \in S_{ℋ} \to A \subseteq ℋ$
2		ococss	$⊢ A \subseteq ℋ \to A \subseteq ⊥ (⊥ (A))$
3	1 2	syl	$⊢ A \in S_{ℋ} \to A \subseteq ⊥ (⊥ (A))$