ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pnfnre GIF version

Theorem pnfnre 7126
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 7063 . . . . . 6 ℂ ∈ V
21uniex 4202 . . . . 5 ℂ ∈ V
3 pwuninel2 5928 . . . . 5 ( ℂ ∈ V → ¬ 𝒫 ℂ ∈ ℂ)
42, 3ax-mp 7 . . . 4 ¬ 𝒫 ℂ ∈ ℂ
5 df-pnf 7121 . . . . 5 +∞ = 𝒫
65eleq1i 2119 . . . 4 (+∞ ∈ ℂ ↔ 𝒫 ℂ ∈ ℂ)
74, 6mtbir 606 . . 3 ¬ +∞ ∈ ℂ
8 recn 7072 . . 3 (+∞ ∈ ℝ → +∞ ∈ ℂ)
97, 8mto 598 . 2 ¬ +∞ ∈ ℝ
109nelir 2317 1 +∞ ∉ ℝ
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wcel 1409  wnel 2314  Vcvv 2574  𝒫 cpw 3387   cuni 3608  cc 6945  cr 6946  +∞cpnf 7116
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-un 4198  ax-cnex 7033  ax-resscn 7034
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-nel 2315  df-rex 2329  df-rab 2332  df-v 2576  df-in 2952  df-ss 2959  df-pw 3389  df-uni 3609  df-pnf 7121
This theorem is referenced by:  renepnf  7132  xrltnr  8802  pnfnlt  8809
  Copyright terms: Public domain W3C validator