Theorem inn0f

Description: A nonempty intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020)

	Ref	Expression
Hypotheses	inn0f.1	$$⊢\underset{\_}{Ⅎ}xA$$
	inn0f.2	$$⊢\underset{\_}{Ⅎ}xB$$
Assertion	inn0f	$$⊢A\cap B\ne \varnothing \leftrightarrow \exists x\in Ax\in B$$

Proof

Step	Hyp	Ref	Expression
1		inn0f.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		inn0f.2	$$⊢\underset{\_}{Ⅎ}xB$$
3		elin	$$⊢x\in \left(A\cap B\right)\leftrightarrow \left(x\in A\wedge x\in B\right)$$
4	3	exbii	$$⊢\exists xx\in \left(A\cap B\right)\leftrightarrow \exists x\left(x\in A\wedge x\in B\right)$$
5	1 2	nfin	$$⊢\underset{\_}{Ⅎ}x\left(A\cap B\right)$$
6	5	n0f	$$⊢A\cap B\ne \varnothing \leftrightarrow \exists xx\in \left(A\cap B\right)$$
7		df-rex	$$⊢\exists x\in Ax\in B\leftrightarrow \exists x\left(x\in A\wedge x\in B\right)$$
8	4 6 7	3bitr4i	$$⊢A\cap B\ne \varnothing \leftrightarrow \exists x\in Ax\in B$$