Theorem xrgepnfd

Description: An extended real greater than or equal to +oo is +oo (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Hypotheses	xrgepnfd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
	xrgepnfd.2	⊢ ( 𝜑 → +∞ ≤ 𝐴 )
Assertion	xrgepnfd	⊢ ( 𝜑 → 𝐴 = +∞ )

Proof

Step	Hyp	Ref	Expression
1		xrgepnfd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
2		xrgepnfd.2	⊢ ( 𝜑 → +∞ ≤ 𝐴 )
3		pnfxr	⊢ +∞ ∈ ℝ*
4	3	a1i	⊢ ( 𝜑 → +∞ ∈ ℝ* )
5		pnfge	⊢ ( 𝐴 ∈ ℝ* → 𝐴 ≤ +∞ )
6	1 5	syl	⊢ ( 𝜑 → 𝐴 ≤ +∞ )
7	1 4 6 2	xrletrid	⊢ ( 𝜑 → 𝐴 = +∞ )