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Theorem pnfnemnf 11300
Description: Plus and minus infinity are different elements of *. (Contributed by NM, 14-Oct-2005.)
Assertion
Ref Expression
pnfnemnf +∞ ≠ -∞

Proof of Theorem pnfnemnf
StepHypRef Expression
1 pnfxr 11299 . . . 4 +∞ ∈ ℝ*
2 pwne 5352 . . . 4 (+∞ ∈ ℝ* → 𝒫 +∞ ≠ +∞)
31, 2ax-mp 5 . . 3 𝒫 +∞ ≠ +∞
43necomi 2992 . 2 +∞ ≠ 𝒫 +∞
5 df-mnf 11282 . 2 -∞ = 𝒫 +∞
64, 5neeqtrri 3011 1 +∞ ≠ -∞
Colors of variables: wff setvar class
Syntax hints:  wcel 2099  wne 2937  𝒫 cpw 4603  +∞cpnf 11276  -∞cmnf 11277  *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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