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Mirrors > Home > ILE Home > Th. List > pnfnemnf | 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 8692 | . . . 4 ⊢ +∞ ∈ ℝ* | |
2 | pwne 3913 | . . . 4 ⊢ (+∞ ∈ ℝ* → 𝒫 +∞ ≠ +∞) | |
3 | 1, 2 | ax-mp 7 | . . 3 ⊢ 𝒫 +∞ ≠ +∞ |
4 | 3 | necomi 2290 | . 2 ⊢ +∞ ≠ 𝒫 +∞ |
5 | df-mnf 7063 | . 2 ⊢ -∞ = 𝒫 +∞ | |
6 | 4, 5 | neeqtrri 2234 | 1 ⊢ +∞ ≠ -∞ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1393 ≠ wne 2204 𝒫 cpw 3359 +∞cpnf 7057 -∞cmnf 7058 ℝ*cxr 7059 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-un 4170 ax-cnex 6975 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-rex 2312 df-rab 2315 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-uni 3581 df-pnf 7062 df-mnf 7063 df-xr 7064 |
This theorem is referenced by: mnfnepnf 8698 xrnemnf 8699 xrltnr 8701 pnfnlt 8708 nltmnf 8709 ngtmnft 8731 |
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