Theorem pnfnemnf

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

	Ref	Expression
Assertion	pnfnemnf	$$⊢\mathrm{+\infty }\ne \mathrm{-\infty }$$

Proof

Step	Hyp	Ref	Expression
1		pnfxr	$$⊢\mathrm{+\infty }\in {ℝ}^{*}$$
2		pwne	$$⊢\mathrm{+\infty }\in {ℝ}^{*}\to 𝒫\mathrm{+\infty }\ne \mathrm{+\infty }$$
3	1 2	ax-mp	$$⊢𝒫\mathrm{+\infty }\ne \mathrm{+\infty }$$
4	3	necomi	$$⊢\mathrm{+\infty }\ne 𝒫\mathrm{+\infty }$$
5		df-mnf	$$⊢\mathrm{-\infty }=𝒫\mathrm{+\infty }$$
6	4 5	neeqtrri	$$⊢\mathrm{+\infty }\ne \mathrm{-\infty }$$