Theorem reim0bi

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

	Ref	Expression
Hypothesis	recl.1	$$⊢A\in ℂ$$
Assertion	reim0bi	$$⊢A\in ℝ\leftrightarrow \Im \left(A\right)=0$$

Proof

Step	Hyp	Ref	Expression
1		recl.1	$$⊢A\in ℂ$$
2		reim0b	$$⊢A\in ℂ\to \left(A\in ℝ\leftrightarrow \Im \left(A\right)=0\right)$$
3	1 2	ax-mp	$$⊢A\in ℝ\leftrightarrow \Im \left(A\right)=0$$