Theorem zabs0b

Description: An integer has an absolute value less than 1 iff it is 0. (Contributed by AV, 21-Nov-2025)

	Ref	Expression
Assertion	zabs0b	⊢ ( 𝑋 ∈ ℤ → ( ( abs ‘ 𝑋 ) < 1 ↔ 𝑋 = 0 ) )

Proof

Step	Hyp	Ref	Expression
1		nn0abscl	⊢ ( 𝑋 ∈ ℤ → ( abs ‘ 𝑋 ) ∈ ℕ0 )
2		nn0lt10b	⊢ ( ( abs ‘ 𝑋 ) ∈ ℕ0 → ( ( abs ‘ 𝑋 ) < 1 ↔ ( abs ‘ 𝑋 ) = 0 ) )
3	1 2	syl	⊢ ( 𝑋 ∈ ℤ → ( ( abs ‘ 𝑋 ) < 1 ↔ ( abs ‘ 𝑋 ) = 0 ) )
4		zcn	⊢ ( 𝑋 ∈ ℤ → 𝑋 ∈ ℂ )
5	4	abs00ad	⊢ ( 𝑋 ∈ ℤ → ( ( abs ‘ 𝑋 ) = 0 ↔ 𝑋 = 0 ) )
6	3 5	bitrd	⊢ ( 𝑋 ∈ ℤ → ( ( abs ‘ 𝑋 ) < 1 ↔ 𝑋 = 0 ) )