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Mirrors > Home > MPE Home > Th. List > pnfnemnf | Structured version Visualization version GIF version |
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 10912 | . . . 4 ⊢ +∞ ∈ ℝ* | |
2 | pwne 5257 | . . . 4 ⊢ (+∞ ∈ ℝ* → 𝒫 +∞ ≠ +∞) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ 𝒫 +∞ ≠ +∞ |
4 | 3 | necomi 2996 | . 2 ⊢ +∞ ≠ 𝒫 +∞ |
5 | df-mnf 10895 | . 2 ⊢ -∞ = 𝒫 +∞ | |
6 | 4, 5 | neeqtrri 3015 | 1 ⊢ +∞ ≠ -∞ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2111 ≠ wne 2941 𝒫 cpw 4528 +∞cpnf 10889 -∞cmnf 10890 ℝ*cxr 10891 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-ext 2709 ax-sep 5207 ax-pow 5273 ax-un 7542 ax-cnex 10810 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-sb 2072 df-clab 2716 df-cleq 2730 df-clel 2817 df-ne 2942 df-rab 3071 df-v 3423 df-un 3886 df-in 3888 df-ss 3898 df-pw 4530 df-sn 4557 df-pr 4559 df-uni 4835 df-pnf 10894 df-mnf 10895 df-xr 10896 |
This theorem is referenced by: mnfnepnf 10914 xnn0nemnf 12198 xrnemnf 12734 xrltnr 12736 pnfnlt 12745 nltmnf 12746 xaddpnf1 12841 xaddnemnf 12851 xmullem2 12880 xadddilem 12909 hashnemnf 13938 xrge0iifhom 31628 esumpr2 31774 |
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