Theorem xrred

Description: An extended real that is neither minus infinity, nor plus infinity, is real. (Contributed by Glauco Siliprandi, 3-Mar-2021)

	Ref	Expression
Hypotheses	xrred.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
	xrred.2	⊢ ( 𝜑 → 𝐴 ≠ -∞ )
	xrred.3	⊢ ( 𝜑 → 𝐴 ≠ +∞ )
Assertion	xrred	⊢ ( 𝜑 → 𝐴 ∈ ℝ )

Proof

Step	Hyp	Ref	Expression
1		xrred.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
2		xrred.2	⊢ ( 𝜑 → 𝐴 ≠ -∞ )
3		xrred.3	⊢ ( 𝜑 → 𝐴 ≠ +∞ )
4	1 2	jca	⊢ ( 𝜑 → ( 𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞ ) )
5		xrnemnf	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞ ) ↔ ( 𝐴 ∈ ℝ ∨ 𝐴 = +∞ ) )
6	4 5	sylib	⊢ ( 𝜑 → ( 𝐴 ∈ ℝ ∨ 𝐴 = +∞ ) )
7	3	neneqd	⊢ ( 𝜑 → ¬ 𝐴 = +∞ )
8		pm2.53	⊢ ( ( 𝐴 ∈ ℝ ∨ 𝐴 = +∞ ) → ( ¬ 𝐴 ∈ ℝ → 𝐴 = +∞ ) )
9	8	con1d	⊢ ( ( 𝐴 ∈ ℝ ∨ 𝐴 = +∞ ) → ( ¬ 𝐴 = +∞ → 𝐴 ∈ ℝ ) )
10	6 7 9	sylc	⊢ ( 𝜑 → 𝐴 ∈ ℝ )