Theorem minel

Description: A minimum element of a class has no elements in common with the class. (Contributed by NM, 22-Jun-1994) (Proof shortened by JJ, 14-Jul-2021)

	Ref	Expression
Assertion	minel	$$⊢\left(A\in B\wedge C\cap B=\varnothing \right)\to \neg A\in C$$

Proof

Step	Hyp	Ref	Expression
1		inelcm	$$⊢\left(A\in C\wedge A\in B\right)\to C\cap B\ne \varnothing$$
2	1	expcom	$$⊢A\in B\to \left(A\in C\to C\cap B\ne \varnothing \right)$$
3	2	necon2bd	$$⊢A\in B\to \left(C\cap B=\varnothing \to \neg A\in C\right)$$
4	3	imp	$$⊢\left(A\in B\wedge C\cap B=\varnothing \right)\to \neg A\in C$$