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Theorem renemnf 7074
Description: No real equals minus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
Assertion
Ref Expression
renemnf (𝐴 ∈ ℝ → 𝐴 ≠ -∞)

Proof of Theorem renemnf
StepHypRef Expression
1 mnfnre 7068 . . . 4 -∞ ∉ ℝ
21neli 2299 . . 3 ¬ -∞ ∈ ℝ
3 eleq1 2100 . . 3 (𝐴 = -∞ → (𝐴 ∈ ℝ ↔ -∞ ∈ ℝ))
42, 3mtbiri 600 . 2 (𝐴 = -∞ → ¬ 𝐴 ∈ ℝ)
54necon2ai 2259 1 (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1243  wcel 1393  wne 2204  cr 6888  -∞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-ne 2206  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:  renemnfd  7077  renfdisj  7079  ltxrlt  7085  xrnemnf  8699  xrlttri3  8718  ngtmnft  8731  xrrebnd  8732  rexneg  8743
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