Theorem ndisj

Description: Express that an intersection is not empty. (Contributed by RP, 16-Apr-2020)

	Ref	Expression
Assertion	ndisj	$$⊢A\cap B\ne \varnothing \leftrightarrow \exists x\left(x\in A\wedge x\in B\right)$$

Proof

Step	Hyp	Ref	Expression
1		n0	$$⊢A\cap B\ne \varnothing \leftrightarrow \exists xx\in \left(A\cap B\right)$$
2		elin	$$⊢x\in \left(A\cap B\right)\leftrightarrow \left(x\in A\wedge x\in B\right)$$
3	2	exbii	$$⊢\exists xx\in \left(A\cap B\right)\leftrightarrow \exists x\left(x\in A\wedge x\in B\right)$$
4	1 3	bitri	$$⊢A\cap B\ne \varnothing \leftrightarrow \exists x\left(x\in A\wedge x\in B\right)$$