Theorem nelcon3d

Description: Contrapositive law deduction for negated membership. (Contributed by AV, 28-Jan-2020)

	Ref	Expression
Hypothesis	nelcon3d.1	|- ( ph -> ( A e. B -> C e. D ) )
Assertion	nelcon3d	|- ( ph -> ( C e/ D -> A e/ B ) )

Proof

Step	Hyp	Ref	Expression
1		nelcon3d.1	|- ( ph -> ( A e. B -> C e. D ) )
2	1	con3d	|- ( ph -> ( -. C e. D -> -. A e. B ) )
3		df-nel	|- ( C e/ D <-> -. C e. D )
4		df-nel	|- ( A e/ B <-> -. A e. B )
5	2 3 4	3imtr4g	|- ( ph -> ( C e/ D -> A e/ B ) )