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	|- ( ph -> A C_ B )
	ssneldd.2	|- ( ph -> -. C e. B )
Assertion	ssneldd	|- ( ph -> -. C e. A )

Proof

Step	Hyp	Ref	Expression
1		ssneld.1	|- ( ph -> A C_ B )
2		ssneldd.2	|- ( ph -> -. C e. B )
3	1	ssneld	|- ( ph -> ( -. C e. B -> -. C e. A ) )
4	2 3	mpd	|- ( ph -> -. C e. A )