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Theorem pnfnre 10032
Description: Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
Assertion
Ref Expression
pnfnre +∞ ∉ ℝ

Proof of Theorem pnfnre
StepHypRef Expression
1 pwuninel 7353 . . . 4 ¬ 𝒫 ℂ ∈ ℂ
2 df-pnf 10027 . . . . 5 +∞ = 𝒫
32eleq1i 2689 . . . 4 (+∞ ∈ ℂ ↔ 𝒫 ℂ ∈ ℂ)
41, 3mtbir 313 . . 3 ¬ +∞ ∈ ℂ
5 recn 9977 . . 3 (+∞ ∈ ℝ → +∞ ∈ ℂ)
64, 5mto 188 . 2 ¬ +∞ ∈ ℝ
76nelir 2896 1 +∞ ∉ ℝ
Colors of variables: wff setvar class
Syntax hints:  wcel 1987  wnel 2893  𝒫 cpw 4135   cuni 4407  cc 9885  cr 9886  +∞cpnf 10022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-resscn 9944
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-nel 2894  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-pw 4137  df-sn 4154  df-pr 4156  df-uni 4408  df-pnf 10027
This theorem is referenced by:  renepnf  10038  ltxrlt  10059  nn0nepnf  11322  xrltnr  11904  pnfnlt  11913  xnn0lenn0nn0  12025  hashclb  13096  hasheq0  13101  pcgcd1  15512  pc2dvds  15514  ramtcl2  15646  odhash3  17919  xrsdsreclblem  19720  pnfnei  20943  iccpnfcnv  22662  i1f0rn  23368  pnfnre2  39115
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