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Theorem pnfnre 6844
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 cnex 6783 . . . . . 6 V
21uniex 4140 . . . . 5 V
3 pwuninel2 5838 . . . . 5 ( V → ¬ 𝒫 ℂ)
42, 3ax-mp 7 . . . 4 ¬ 𝒫
5 df-pnf 6839 . . . . 5 +∞ = 𝒫
65eleq1i 2100 . . . 4 (+∞ ℂ ↔ 𝒫 ℂ)
74, 6mtbir 595 . . 3 ¬ +∞
8 recn 6792 . . 3 (+∞ ℝ → +∞ ℂ)
97, 8mto 587 . 2 ¬ +∞
109nelir 2294 1 +∞ ∉ ℝ
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wcel 1390  wnel 2202  Vcvv 2551  𝒫 cpw 3351   cuni 3571  cc 6689  cr 6690  +∞cpnf 6834
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-un 4136  ax-cnex 6754  ax-resscn 6755
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-nel 2204  df-rex 2306  df-rab 2309  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353  df-uni 3572  df-pnf 6839
This theorem is referenced by:  renepnf  6850  xrltnr  8451  pnfnlt  8458
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