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Theorem pnfnre 7466
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 7403 . . . . . 6 ℂ ∈ V
21uniex 4237 . . . . 5 ℂ ∈ V
3 pwuninel2 5995 . . . . 5 ( ℂ ∈ V → ¬ 𝒫 ℂ ∈ ℂ)
42, 3ax-mp 7 . . . 4 ¬ 𝒫 ℂ ∈ ℂ
5 df-pnf 7461 . . . . 5 +∞ = 𝒫
65eleq1i 2150 . . . 4 (+∞ ∈ ℂ ↔ 𝒫 ℂ ∈ ℂ)
74, 6mtbir 629 . . 3 ¬ +∞ ∈ ℂ
8 recn 7412 . . 3 (+∞ ∈ ℝ → +∞ ∈ ℂ)
97, 8mto 621 . 2 ¬ +∞ ∈ ℝ
109nelir 2349 1 +∞ ∉ ℝ
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wcel 1436  wnel 2346  Vcvv 2615  𝒫 cpw 3415   cuni 3636  cc 7285  cr 7286  +∞cpnf 7456
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3931  ax-un 4233  ax-cnex 7373  ax-resscn 7374
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-nel 2347  df-rex 2361  df-rab 2364  df-v 2617  df-in 2994  df-ss 3001  df-pw 3417  df-uni 3637  df-pnf 7461
This theorem is referenced by:  renepnf  7472  nn0nepnf  8670  xrltnr  9175  pnfnlt  9182  inftonninf  9768
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