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Theorem renfdisj 7836
Description: The reals and the infinities are disjoint. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
Assertion
Ref Expression
renfdisj (ℝ ∩ {+∞, -∞}) = ∅

Proof of Theorem renfdisj
StepHypRef Expression
1 disj 3411 . 2 ((ℝ ∩ {+∞, -∞}) = ∅ ↔ ∀𝑥 ∈ ℝ ¬ 𝑥 ∈ {+∞, -∞})
2 vex 2689 . . . . 5 𝑥 ∈ V
32elpr 3548 . . . 4 (𝑥 ∈ {+∞, -∞} ↔ (𝑥 = +∞ ∨ 𝑥 = -∞))
4 renepnf 7825 . . . . . 6 (𝑥 ∈ ℝ → 𝑥 ≠ +∞)
54necon2bi 2363 . . . . 5 (𝑥 = +∞ → ¬ 𝑥 ∈ ℝ)
6 renemnf 7826 . . . . . 6 (𝑥 ∈ ℝ → 𝑥 ≠ -∞)
76necon2bi 2363 . . . . 5 (𝑥 = -∞ → ¬ 𝑥 ∈ ℝ)
85, 7jaoi 705 . . . 4 ((𝑥 = +∞ ∨ 𝑥 = -∞) → ¬ 𝑥 ∈ ℝ)
93, 8sylbi 120 . . 3 (𝑥 ∈ {+∞, -∞} → ¬ 𝑥 ∈ ℝ)
109con2i 616 . 2 (𝑥 ∈ ℝ → ¬ 𝑥 ∈ {+∞, -∞})
111, 10mprgbir 2490 1 (ℝ ∩ {+∞, -∞}) = ∅
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wo 697   = wceq 1331  wcel 1480  cin 3070  c0 3363  {cpr 3528  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-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-un 4355  ax-setind 4452  ax-cnex 7723  ax-resscn 7724
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-uni 3737  df-pnf 7814  df-mnf 7815
This theorem is referenced by: (None)
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