Theorem pnfnemnf

Description: Plus and minus infinity are different elements of RR* . (Contributed by NM, 14-Oct-2005)

	Ref	Expression
Assertion	pnfnemnf	⊢ +∞ ≠ -∞

Proof

Step	Hyp	Ref	Expression
1		pnfxr	⊢ +∞ ∈ ℝ*
2		pwne	⊢ ( +∞ ∈ ℝ* → 𝒫 +∞ ≠ +∞ )
3	1 2	ax-mp	⊢ 𝒫 +∞ ≠ +∞
4	3	necomi	⊢ +∞ ≠ 𝒫 +∞
5		df-mnf	⊢ -∞ = 𝒫 +∞
6	4 5	neeqtrri	⊢ +∞ ≠ -∞