Theorem rerebi

Description: A real number equals its real part. Proposition 10-3.4(f) of Gleason p. 133. (Contributed by NM, 27-Oct-1999)

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

Proof

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