chsscon1

Description: Hilbert lattice contraposition law. (Contributed by NM, 21-Jun-2004) (New usage is discouraged.)

		Ref	Expression
	Assertion	chsscon1	$⊢ (A \in C_{ℋ} \land B \in C_{ℋ}) \to (⊥ (A) \subseteq B \leftrightarrow ⊥ (B) \subseteq A)$

Step	Hyp	Ref	Expression
1		choccl	$⊢ A \in C_{ℋ} \to ⊥ (A) \in C_{ℋ}$
2		chsscon3	$⊢ (⊥ (A) \in C_{ℋ} \land B \in C_{ℋ}) \to (⊥ (A) \subseteq B \leftrightarrow ⊥ (B) \subseteq ⊥ (⊥ (A)))$
3	1 2	sylan	$⊢ (A \in C_{ℋ} \land B \in C_{ℋ}) \to (⊥ (A) \subseteq B \leftrightarrow ⊥ (B) \subseteq ⊥ (⊥ (A)))$
4		ococ	$⊢ A \in C_{ℋ} \to ⊥ (⊥ (A)) = A$
5	4	adantr	$⊢ (A \in C_{ℋ} \land B \in C_{ℋ}) \to ⊥ (⊥ (A)) = A$
6	5	sseq2d	$⊢ (A \in C_{ℋ} \land B \in C_{ℋ}) \to (⊥ (B) \subseteq ⊥ (⊥ (A)) \leftrightarrow ⊥ (B) \subseteq A)$
7	3 6	bitrd	$⊢ (A \in C_{ℋ} \land B \in C_{ℋ}) \to (⊥ (A) \subseteq B \leftrightarrow ⊥ (B) \subseteq A)$

Metamath Proof Explorer