Theorem n0rex

Description: There is an element in a nonempty class which is an element of the class. (Contributed by AV, 17-Dec-2020)

	Ref	Expression
Assertion	n0rex	$$⊢A\ne \varnothing \to \exists x\in Ax\in A$$

Proof

Step	Hyp	Ref	Expression
1		id	$$⊢x\in A\to x\in A$$
2	1	ancli	$$⊢x\in A\to \left(x\in A\wedge x\in A\right)$$
3	2	eximi	$$⊢\exists xx\in A\to \exists x\left(x\in A\wedge x\in A\right)$$
4		n0	$$⊢A\ne \varnothing \leftrightarrow \exists xx\in A$$
5		df-rex	$$⊢\exists x\in Ax\in A\leftrightarrow \exists x\left(x\in A\wedge x\in A\right)$$
6	3 4 5	3imtr4i	$$⊢A\ne \varnothing \to \exists x\in Ax\in A$$