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Metamath Proof Explorer

Theorem ssneldd

Description: If an element is not in a class, it is also not in a subclass of that class. Deduction form. (Contributed by David Moews, 1-May-2017)

	Ref	Expression
Hypotheses	ssneld.1	$⊢ φ \to A \subseteq B$
	ssneldd.2	$⊢ φ \to \neg C \in B$
Assertion	ssneldd	$⊢ φ \to \neg C \in A$

Proof

Step	Hyp	Ref	Expression
1		ssneld.1	$⊢ φ \to A \subseteq B$
2		ssneldd.2	$⊢ φ \to \neg C \in B$
3	1	ssneld	$⊢ φ \to (\neg C \in B \to \neg C \in A)$
4	2 3	mpd	$⊢ φ \to \neg C \in A$