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Theorem mnfnre 7474
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 7410 . . . . 5 ℂ ∈ V
2 2pwuninelg 6002 . . . . 5 (ℂ ∈ V → ¬ 𝒫 𝒫 ℂ ∈ ℂ)
31, 2ax-mp 7 . . . 4 ¬ 𝒫 𝒫 ℂ ∈ ℂ
4 df-mnf 7469 . . . . . 6 -∞ = 𝒫 +∞
5 df-pnf 7468 . . . . . . 7 +∞ = 𝒫
65pweqi 3419 . . . . . 6 𝒫 +∞ = 𝒫 𝒫
74, 6eqtri 2105 . . . . 5 -∞ = 𝒫 𝒫
87eleq1i 2150 . . . 4 (-∞ ∈ ℂ ↔ 𝒫 𝒫 ℂ ∈ ℂ)
93, 8mtbir 629 . . 3 ¬ -∞ ∈ ℂ
10 recn 7419 . . 3 (-∞ ∈ ℝ → -∞ ∈ ℂ)
119, 10mto 621 . 2 ¬ -∞ ∈ ℝ
1211nelir 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 7292  cr 7293  +∞cpnf 7463  -∞cmnf 7464
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-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-setind 4326  ax-cnex 7380  ax-resscn 7381
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-nel 2347  df-ral 2360  df-v 2617  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-uni 3637  df-pnf 7468  df-mnf 7469
This theorem is referenced by:  renemnf  7480  xrltnr  9182  nltmnf  9190
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