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Theorem renepnf 7131
Description: No (finite) real equals plus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
Assertion
Ref Expression
renepnf (𝐴 ∈ ℝ → 𝐴 ≠ +∞)

Proof of Theorem renepnf
StepHypRef Expression
1 pnfnre 7125 . . . 4 +∞ ∉ ℝ
21neli 2316 . . 3 ¬ +∞ ∈ ℝ
3 eleq1 2116 . . 3 (𝐴 = +∞ → (𝐴 ∈ ℝ ↔ +∞ ∈ ℝ))
42, 3mtbiri 610 . 2 (𝐴 = +∞ → ¬ 𝐴 ∈ ℝ)
54necon2ai 2274 1 (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  wcel 1409  wne 2220  cr 6945  +∞cpnf 7115
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-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-un 4197  ax-cnex 7032  ax-resscn 7033
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-rex 2329  df-rab 2332  df-v 2576  df-in 2951  df-ss 2958  df-pw 3388  df-uni 3608  df-pnf 7120
This theorem is referenced by:  renepnfd  7134  renfdisj  7137  ltxrlt  7143  xrnepnf  8800  xrlttri3  8818  nltpnft  8830  xrrebnd  8832  rexneg  8843
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