Theorem eldifbd

Description: If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif . (Contributed by David Moews, 1-May-2017)

	Ref	Expression
Hypothesis	eldifbd.1	$$⊢\phi \to A\in \left(B\setminus C\right)$$
Assertion	eldifbd	$$⊢\phi \to \neg A\in C$$

Proof

Step	Hyp	Ref	Expression
1		eldifbd.1	$$⊢\phi \to A\in \left(B\setminus C\right)$$
2		eldif	$$⊢A\in \left(B\setminus C\right)\leftrightarrow \left(A\in B\wedge \neg A\in C\right)$$
3	1 2	sylib	$$⊢\phi \to \left(A\in B\wedge \neg A\in C\right)$$
4	3	simprd	$$⊢\phi \to \neg A\in C$$