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| Description: Plus and minus infinity are different elements of ℝ*. (Contributed by NM, 14-Oct-2005.) | 
| Ref | Expression | 
|---|---|
| pnfnemnf | ⊢ +∞ ≠ -∞ | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | pnfxr 11315 | . . . 4 ⊢ +∞ ∈ ℝ* | |
| 2 | pwne 5353 | . . . 4 ⊢ (+∞ ∈ ℝ* → 𝒫 +∞ ≠ +∞) | |
| 3 | 1, 2 | ax-mp 5 | . . 3 ⊢ 𝒫 +∞ ≠ +∞ | 
| 4 | 3 | necomi 2995 | . 2 ⊢ +∞ ≠ 𝒫 +∞ | 
| 5 | df-mnf 11298 | . 2 ⊢ -∞ = 𝒫 +∞ | |
| 6 | 4, 5 | neeqtrri 3014 | 1 ⊢ +∞ ≠ -∞ | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∈ wcel 2108 ≠ wne 2940 𝒫 cpw 4600 +∞cpnf 11292 -∞cmnf 11293 ℝ*cxr 11294 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-pow 5365 ax-un 7755 ax-cnex 11211 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-rab 3437 df-v 3482 df-un 3956 df-in 3958 df-ss 3968 df-pw 4602 df-sn 4627 df-pr 4629 df-uni 4908 df-pnf 11297 df-mnf 11298 df-xr 11299 | 
| This theorem is referenced by: mnfnepnf 11317 xnn0nemnf 12610 xrnemnf 13159 xrltnr 13161 pnfnlt 13170 nltmnf 13171 xaddpnf1 13268 xaddnemnf 13278 xmullem2 13307 xadddilem 13336 hashnemnf 14383 xrge0iifhom 33936 esumpr2 34068 | 
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