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 ∈ ℂ
Assertion	rerebi	⊢ ( 𝐴 ∈ ℝ ↔ ( ℜ ‘ 𝐴 ) = 𝐴 )

Proof

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