Theorem minel

Description: A minimum element of a class has no elements in common with the class. (Contributed by NM, 22-Jun-1994) (Proof shortened by JJ, 14-Jul-2021)

	Ref	Expression
Assertion	minel	⊢ ( ( 𝐴 ∈ 𝐵 ∧ ( 𝐶 ∩ 𝐵 ) = ∅ ) → ¬ 𝐴 ∈ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		inelcm	⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐴 ∈ 𝐵 ) → ( 𝐶 ∩ 𝐵 ) ≠ ∅ )
2	1	expcom	⊢ ( 𝐴 ∈ 𝐵 → ( 𝐴 ∈ 𝐶 → ( 𝐶 ∩ 𝐵 ) ≠ ∅ ) )
3	2	necon2bd	⊢ ( 𝐴 ∈ 𝐵 → ( ( 𝐶 ∩ 𝐵 ) = ∅ → ¬ 𝐴 ∈ 𝐶 ) )
4	3	imp	⊢ ( ( 𝐴 ∈ 𝐵 ∧ ( 𝐶 ∩ 𝐵 ) = ∅ ) → ¬ 𝐴 ∈ 𝐶 )