Theorem inn0

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

	Ref	Expression
Assertion	inn0	⊢ ( ( 𝐴 ∩ 𝐵 ) ≠ ∅ ↔ ∃ 𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		nfcv	⊢ Ⅎ 𝑥 𝐴
2		nfcv	⊢ Ⅎ 𝑥 𝐵
3	1 2	inn0f	⊢ ( ( 𝐴 ∩ 𝐵 ) ≠ ∅ ↔ ∃ 𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 )