Theorem ndisj

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

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

Proof

Step	Hyp	Ref	Expression
1		n0	⊢ ( ( 𝐴 ∩ 𝐵 ) ≠ ∅ ↔ ∃ 𝑥 𝑥 ∈ ( 𝐴 ∩ 𝐵 ) )
2		elin	⊢ ( 𝑥 ∈ ( 𝐴 ∩ 𝐵 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) )
3	2	exbii	⊢ ( ∃ 𝑥 𝑥 ∈ ( 𝐴 ∩ 𝐵 ) ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) )
4	1 3	bitri	⊢ ( ( 𝐴 ∩ 𝐵 ) ≠ ∅ ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) )