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	$$⊢\phi \to A\subseteq B$$
	ssneldd.2	$$⊢\phi \to \neg C\in B$$
Assertion	ssneldd	$$⊢\phi \to \neg C\in A$$

Proof

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