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Theorem mnfnre 7068
Description: Minus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
Assertion
Ref Expression
mnfnre -∞ ∉ ℝ

Proof of Theorem mnfnre
StepHypRef Expression
1 cnex 7005 . . . . 5 ℂ ∈ V
2 2pwuninelg 5898 . . . . 5 (ℂ ∈ V → ¬ 𝒫 𝒫 ℂ ∈ ℂ)
31, 2ax-mp 7 . . . 4 ¬ 𝒫 𝒫 ℂ ∈ ℂ
4 df-mnf 7063 . . . . . 6 -∞ = 𝒫 +∞
5 df-pnf 7062 . . . . . . 7 +∞ = 𝒫
65pweqi 3363 . . . . . 6 𝒫 +∞ = 𝒫 𝒫
74, 6eqtri 2060 . . . . 5 -∞ = 𝒫 𝒫
87eleq1i 2103 . . . 4 (-∞ ∈ ℂ ↔ 𝒫 𝒫 ℂ ∈ ℂ)
93, 8mtbir 596 . . 3 ¬ -∞ ∈ ℂ
10 recn 7014 . . 3 (-∞ ∈ ℝ → -∞ ∈ ℂ)
119, 10mto 588 . 2 ¬ -∞ ∈ ℝ
1211nelir 2300 1 -∞ ∉ ℝ
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wcel 1393  wnel 2205  Vcvv 2557  𝒫 cpw 3359   cuni 3580  cc 6887  cr 6888  +∞cpnf 7057  -∞cmnf 7058
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-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-setind 4262  ax-cnex 6975  ax-resscn 6976
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-nel 2207  df-ral 2311  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-pnf 7062  df-mnf 7063
This theorem is referenced by:  renemnf  7074  xrltnr  8701  nltmnf  8709
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