Theorem inelcm

Description: The intersection of classes with a common member is nonempty. (Contributed by NM, 7-Apr-1994)

	Ref	Expression
Assertion	inelcm	⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶 ) → ( 𝐵 ∩ 𝐶 ) ≠ ∅ )

Proof

Step	Hyp	Ref	Expression
1		elin	⊢ ( 𝐴 ∈ ( 𝐵 ∩ 𝐶 ) ↔ ( 𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶 ) )
2		ne0i	⊢ ( 𝐴 ∈ ( 𝐵 ∩ 𝐶 ) → ( 𝐵 ∩ 𝐶 ) ≠ ∅ )
3	1 2	sylbir	⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶 ) → ( 𝐵 ∩ 𝐶 ) ≠ ∅ )