Theorem fzo0

Description: Half-open sets with equal endpoints are empty. (Contributed by Stefan O'Rear, 15-Aug-2015) (Revised by Mario Carneiro, 29-Sep-2015)

	Ref	Expression
Assertion	fzo0	⊢ ( 𝐴 ..^ 𝐴 ) = ∅

Proof

Step	Hyp	Ref	Expression
1		fzonel	⊢ ¬ 𝐴 ∈ ( 𝐴 ..^ 𝐴 )
2		fzon0	⊢ ( ( 𝐴 ..^ 𝐴 ) ≠ ∅ ↔ 𝐴 ∈ ( 𝐴 ..^ 𝐴 ) )
3	2	necon1bbii	⊢ ( ¬ 𝐴 ∈ ( 𝐴 ..^ 𝐴 ) ↔ ( 𝐴 ..^ 𝐴 ) = ∅ )
4	1 3	mpbi	⊢ ( 𝐴 ..^ 𝐴 ) = ∅