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Theorem renfdisj 7607
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 3335 . 2 ((ℝ ∩ {+∞, -∞}) = ∅ ↔ ∀𝑥 ∈ ℝ ¬ 𝑥 ∈ {+∞, -∞})
2 vex 2623 . . . . 5 𝑥 ∈ V
32elpr 3471 . . . 4 (𝑥 ∈ {+∞, -∞} ↔ (𝑥 = +∞ ∨ 𝑥 = -∞))
4 renepnf 7596 . . . . . 6 (𝑥 ∈ ℝ → 𝑥 ≠ +∞)
54necon2bi 2311 . . . . 5 (𝑥 = +∞ → ¬ 𝑥 ∈ ℝ)
6 renemnf 7597 . . . . . 6 (𝑥 ∈ ℝ → 𝑥 ≠ -∞)
76necon2bi 2311 . . . . 5 (𝑥 = -∞ → ¬ 𝑥 ∈ ℝ)
85, 7jaoi 672 . . . 4 ((𝑥 = +∞ ∨ 𝑥 = -∞) → ¬ 𝑥 ∈ ℝ)
93, 8sylbi 120 . . 3 (𝑥 ∈ {+∞, -∞} → ¬ 𝑥 ∈ ℝ)
109con2i 593 . 2 (𝑥 ∈ ℝ → ¬ 𝑥 ∈ {+∞, -∞})
111, 10mprgbir 2434 1 (ℝ ∩ {+∞, -∞}) = ∅
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wo 665   = wceq 1290  wcel 1439  cin 2999  c0 3287  {cpr 3451  cr 7410  +∞cpnf 7580  -∞cmnf 7581
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-un 4269  ax-setind 4366  ax-cnex 7497  ax-resscn 7498
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-rab 2369  df-v 2622  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-uni 3660  df-pnf 7585  df-mnf 7586
This theorem is referenced by: (None)
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