Theorem 0pssin

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

	Ref	Expression
Assertion	0pssin	$$⊢\varnothing \subset A\cap B\leftrightarrow \exists x\left(x\in A\wedge x\in B\right)$$

Proof

Step	Hyp	Ref	Expression
1		0pss	$$⊢\varnothing \subset A\cap B\leftrightarrow A\cap B\ne \varnothing$$
2		ndisj	$$⊢A\cap B\ne \varnothing \leftrightarrow \exists x\left(x\in A\wedge x\in B\right)$$
3	1 2	bitri	$$⊢\varnothing \subset A\cap B\leftrightarrow \exists x\left(x\in A\wedge x\in B\right)$$