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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 11316 | . . . 4 ⊢ +∞ ∈ ℝ* | |
| 2 | pwne 5352 | . . . 4 ⊢ (+∞ ∈ ℝ* → 𝒫 +∞ ≠ +∞) | |
| 3 | 1, 2 | ax-mp 5 | . . 3 ⊢ 𝒫 +∞ ≠ +∞ | 
| 4 | 3 | necomi 2994 | . 2 ⊢ +∞ ≠ 𝒫 +∞ | 
| 5 | df-mnf 11299 | . 2 ⊢ -∞ = 𝒫 +∞ | |
| 6 | 4, 5 | neeqtrri 3013 | 1 ⊢ +∞ ≠ -∞ | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∈ wcel 2107 ≠ wne 2939 𝒫 cpw 4599 +∞cpnf 11293 -∞cmnf 11294 ℝ*cxr 11295 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-pow 5364 ax-un 7756 ax-cnex 11212 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-rab 3436 df-v 3481 df-un 3955 df-in 3957 df-ss 3967 df-pw 4601 df-sn 4626 df-pr 4628 df-uni 4907 df-pnf 11298 df-mnf 11299 df-xr 11300 | 
| This theorem is referenced by: mnfnepnf 11318 xnn0nemnf 12612 xrnemnf 13160 xrltnr 13162 pnfnlt 13171 nltmnf 13172 xaddpnf1 13269 xaddnemnf 13279 xmullem2 13308 xadddilem 13337 hashnemnf 14384 xrge0iifhom 33937 esumpr2 34069 | 
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