Theorem xrnmnfpnf

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

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

Proof

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