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Metamath Proof Explorer

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