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Theorem renfdisj 7291
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 3308 . 2 ((ℝ ∩ {+∞, -∞}) = ∅ ↔ ∀𝑥 ∈ ℝ ¬ 𝑥 ∈ {+∞, -∞})
2 vex 2613 . . . . 5 𝑥 ∈ V
32elpr 3437 . . . 4 (𝑥 ∈ {+∞, -∞} ↔ (𝑥 = +∞ ∨ 𝑥 = -∞))
4 renepnf 7280 . . . . . 6 (𝑥 ∈ ℝ → 𝑥 ≠ +∞)
54necon2bi 2304 . . . . 5 (𝑥 = +∞ → ¬ 𝑥 ∈ ℝ)
6 renemnf 7281 . . . . . 6 (𝑥 ∈ ℝ → 𝑥 ≠ -∞)
76necon2bi 2304 . . . . 5 (𝑥 = -∞ → ¬ 𝑥 ∈ ℝ)
85, 7jaoi 669 . . . 4 ((𝑥 = +∞ ∨ 𝑥 = -∞) → ¬ 𝑥 ∈ ℝ)
93, 8sylbi 119 . . 3 (𝑥 ∈ {+∞, -∞} → ¬ 𝑥 ∈ ℝ)
109con2i 590 . 2 (𝑥 ∈ ℝ → ¬ 𝑥 ∈ {+∞, -∞})
111, 10mprgbir 2426 1 (ℝ ∩ {+∞, -∞}) = ∅
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wo 662   = wceq 1285  wcel 1434  cin 2981  c0 3267  {cpr 3417  cr 7094  +∞cpnf 7264  -∞cmnf 7265
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-un 4216  ax-setind 4308  ax-cnex 7181  ax-resscn 7182
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-uni 3622  df-pnf 7269  df-mnf 7270
This theorem is referenced by: (None)
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