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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 10417 | . . . 4 ⊢ +∞ ∈ ℝ* | |
2 | pwne 5055 | . . . 4 ⊢ (+∞ ∈ ℝ* → 𝒫 +∞ ≠ +∞) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ 𝒫 +∞ ≠ +∞ |
4 | 3 | necomi 3053 | . 2 ⊢ +∞ ≠ 𝒫 +∞ |
5 | df-mnf 10401 | . 2 ⊢ -∞ = 𝒫 +∞ | |
6 | 4, 5 | neeqtrri 3072 | 1 ⊢ +∞ ≠ -∞ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2164 ≠ wne 2999 𝒫 cpw 4380 +∞cpnf 10395 -∞cmnf 10396 ℝ*cxr 10397 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-pow 5067 ax-un 7214 ax-cnex 10315 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-rex 3123 df-rab 3126 df-v 3416 df-un 3803 df-in 3805 df-ss 3812 df-pw 4382 df-sn 4400 df-pr 4402 df-uni 4661 df-pnf 10400 df-mnf 10401 df-xr 10402 |
This theorem is referenced by: mnfnepnf 10420 xnn0nemnf 11708 xrnemnf 12244 xrltnr 12246 pnfnlt 12255 nltmnf 12256 xaddpnf1 12352 xaddnemnf 12362 xmullem2 12390 xadddilem 12419 hashnemnf 13431 xrge0iifhom 30524 esumpr2 30670 |
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