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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 11299 | . . . 4 ⊢ +∞ ∈ ℝ* | |
2 | pwne 5352 | . . . 4 ⊢ (+∞ ∈ ℝ* → 𝒫 +∞ ≠ +∞) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ 𝒫 +∞ ≠ +∞ |
4 | 3 | necomi 2992 | . 2 ⊢ +∞ ≠ 𝒫 +∞ |
5 | df-mnf 11282 | . 2 ⊢ -∞ = 𝒫 +∞ | |
6 | 4, 5 | neeqtrri 3011 | 1 ⊢ +∞ ≠ -∞ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2099 ≠ wne 2937 𝒫 cpw 4603 +∞cpnf 11276 -∞cmnf 11277 ℝ*cxr 11278 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5299 ax-pow 5365 ax-un 7740 ax-cnex 11195 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-rab 3430 df-v 3473 df-un 3952 df-in 3954 df-ss 3964 df-pw 4605 df-sn 4630 df-pr 4632 df-uni 4909 df-pnf 11281 df-mnf 11282 df-xr 11283 |
This theorem is referenced by: mnfnepnf 11301 xnn0nemnf 12586 xrnemnf 13130 xrltnr 13132 pnfnlt 13141 nltmnf 13142 xaddpnf1 13238 xaddnemnf 13248 xmullem2 13277 xadddilem 13306 hashnemnf 14336 xrge0iifhom 33538 esumpr2 33686 |
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