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Theorem mnfnre 7126
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 7062 . . . . 5 ℂ ∈ V
2 2pwuninelg 5928 . . . . 5 (ℂ ∈ V → ¬ 𝒫 𝒫 ℂ ∈ ℂ)
31, 2ax-mp 7 . . . 4 ¬ 𝒫 𝒫 ℂ ∈ ℂ
4 df-mnf 7121 . . . . . 6 -∞ = 𝒫 +∞
5 df-pnf 7120 . . . . . . 7 +∞ = 𝒫
65pweqi 3390 . . . . . 6 𝒫 +∞ = 𝒫 𝒫
74, 6eqtri 2076 . . . . 5 -∞ = 𝒫 𝒫
87eleq1i 2119 . . . 4 (-∞ ∈ ℂ ↔ 𝒫 𝒫 ℂ ∈ ℂ)
93, 8mtbir 606 . . 3 ¬ -∞ ∈ ℂ
10 recn 7071 . . 3 (-∞ ∈ ℝ → -∞ ∈ ℂ)
119, 10mto 598 . 2 ¬ -∞ ∈ ℝ
1211nelir 2317 1 -∞ ∉ ℝ
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wcel 1409  wnel 2314  Vcvv 2574  𝒫 cpw 3386   cuni 3607  cc 6944  cr 6945  +∞cpnf 7115  -∞cmnf 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-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-setind 4289  ax-cnex 7032  ax-resscn 7033
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-nel 2315  df-ral 2328  df-v 2576  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-uni 3608  df-pnf 7120  df-mnf 7121
This theorem is referenced by:  renemnf  7132  xrltnr  8801  nltmnf  8809
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