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Theorem pnfnre 7067
 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 7005 . . . . . 6 ℂ ∈ V
21uniex 4174 . . . . 5 ℂ ∈ V
3 pwuninel2 5897 . . . . 5 ( ℂ ∈ V → ¬ 𝒫 ℂ ∈ ℂ)
42, 3ax-mp 7 . . . 4 ¬ 𝒫 ℂ ∈ ℂ
5 df-pnf 7062 . . . . 5 +∞ = 𝒫
65eleq1i 2103 . . . 4 (+∞ ∈ ℂ ↔ 𝒫 ℂ ∈ ℂ)
74, 6mtbir 596 . . 3 ¬ +∞ ∈ ℂ
8 recn 7014 . . 3 (+∞ ∈ ℝ → +∞ ∈ ℂ)
97, 8mto 588 . 2 ¬ +∞ ∈ ℝ
109nelir 2300 1 +∞ ∉ ℝ
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∈ wcel 1393   ∉ wnel 2205  Vcvv 2557  𝒫 cpw 3359  ∪ cuni 3580  ℂcc 6887  ℝcr 6888  +∞cpnf 7057 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-un 4170  ax-cnex 6975  ax-resscn 6976 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-nel 2207  df-rex 2312  df-rab 2315  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361  df-uni 3581  df-pnf 7062 This theorem is referenced by:  renepnf  7073  xrltnr  8701  pnfnlt  8708
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