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Metamath Proof Explorer

Theorem chnlen0

Description: A Hilbert lattice element that is not a subset of another is nonzero. (Contributed by NM, 30-Jun-2004) (New usage is discouraged.)

		Ref	Expression
	Assertion	chnlen0	$⊢ B \in C_{ℋ} \to (\neg A \subseteq B \to \neg A = 0_{ℋ})$

Step	Hyp	Ref	Expression
1		ch0le	$⊢ B \in C_{ℋ} \to 0_{ℋ} \subseteq B$
2		sseq1	$⊢ A = 0_{ℋ} \to (A \subseteq B \leftrightarrow 0_{ℋ} \subseteq B)$
3	1 2	syl5ibrcom	$⊢ B \in C_{ℋ} \to (A = 0_{ℋ} \to A \subseteq B)$
4	3	con3d	$⊢ B \in C_{ℋ} \to (\neg A \subseteq B \to \neg A = 0_{ℋ})$