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Theorem mnfnre 7820
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 7756 . . . . 5 ℂ ∈ V
2 2pwuninelg 6180 . . . . 5 (ℂ ∈ V → ¬ 𝒫 𝒫 ℂ ∈ ℂ)
31, 2ax-mp 5 . . . 4 ¬ 𝒫 𝒫 ℂ ∈ ℂ
4 df-mnf 7815 . . . . . 6 -∞ = 𝒫 +∞
5 df-pnf 7814 . . . . . . 7 +∞ = 𝒫
65pweqi 3514 . . . . . 6 𝒫 +∞ = 𝒫 𝒫
74, 6eqtri 2160 . . . . 5 -∞ = 𝒫 𝒫
87eleq1i 2205 . . . 4 (-∞ ∈ ℂ ↔ 𝒫 𝒫 ℂ ∈ ℂ)
93, 8mtbir 660 . . 3 ¬ -∞ ∈ ℂ
10 recn 7765 . . 3 (-∞ ∈ ℝ → -∞ ∈ ℂ)
119, 10mto 651 . 2 ¬ -∞ ∈ ℝ
1211nelir 2406 1 -∞ ∉ ℝ
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wcel 1480  wnel 2403  Vcvv 2686  𝒫 cpw 3510   cuni 3736  cc 7630  cr 7631  +∞cpnf 7809  -∞cmnf 7810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-setind 4452  ax-cnex 7723  ax-resscn 7724
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-nel 2404  df-ral 2421  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-uni 3737  df-pnf 7814  df-mnf 7815
This theorem is referenced by:  renemnf  7826  xrltnr  9578  nltmnf  9586
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