Theorem ceilidz

Description: A real number equals its ceiling iff it is an integer. (Contributed by AV, 30-Nov-2018)

	Ref	Expression
Assertion	ceilidz	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 ∈ ℤ ↔ ( ⌈ ‘ 𝐴 ) = 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		ceilid	⊢ ( 𝐴 ∈ ℤ → ( ⌈ ‘ 𝐴 ) = 𝐴 )
2		ceilcl	⊢ ( 𝐴 ∈ ℝ → ( ⌈ ‘ 𝐴 ) ∈ ℤ )
3		eleq1	⊢ ( ( ⌈ ‘ 𝐴 ) = 𝐴 → ( ( ⌈ ‘ 𝐴 ) ∈ ℤ ↔ 𝐴 ∈ ℤ ) )
4	2 3	syl5ibcom	⊢ ( 𝐴 ∈ ℝ → ( ( ⌈ ‘ 𝐴 ) = 𝐴 → 𝐴 ∈ ℤ ) )
5	1 4	impbid2	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 ∈ ℤ ↔ ( ⌈ ‘ 𝐴 ) = 𝐴 ) )