Theorem reim0bi

Description: A number is real iff its imaginary part is 0. (Contributed by NM, 29-May-1999)

	Ref	Expression
Hypothesis	recl.1	⊢ 𝐴 ∈ ℂ
Assertion	reim0bi	⊢ ( 𝐴 ∈ ℝ ↔ ( ℑ ‘ 𝐴 ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		recl.1	⊢ 𝐴 ∈ ℂ
2		reim0b	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ∈ ℝ ↔ ( ℑ ‘ 𝐴 ) = 0 ) )
3	1 2	ax-mp	⊢ ( 𝐴 ∈ ℝ ↔ ( ℑ ‘ 𝐴 ) = 0 )