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Theorem renfdisj 7079
 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 3268 . 2 ((ℝ ∩ {+∞, -∞}) = ∅ ↔ ∀𝑥 ∈ ℝ ¬ 𝑥 ∈ {+∞, -∞})
2 vex 2560 . . . . 5 𝑥 ∈ V
32elpr 3396 . . . 4 (𝑥 ∈ {+∞, -∞} ↔ (𝑥 = +∞ ∨ 𝑥 = -∞))
4 renepnf 7073 . . . . . 6 (𝑥 ∈ ℝ → 𝑥 ≠ +∞)
54necon2bi 2260 . . . . 5 (𝑥 = +∞ → ¬ 𝑥 ∈ ℝ)
6 renemnf 7074 . . . . . 6 (𝑥 ∈ ℝ → 𝑥 ≠ -∞)
76necon2bi 2260 . . . . 5 (𝑥 = -∞ → ¬ 𝑥 ∈ ℝ)
85, 7jaoi 636 . . . 4 ((𝑥 = +∞ ∨ 𝑥 = -∞) → ¬ 𝑥 ∈ ℝ)
93, 8sylbi 114 . . 3 (𝑥 ∈ {+∞, -∞} → ¬ 𝑥 ∈ ℝ)
109con2i 557 . 2 (𝑥 ∈ ℝ → ¬ 𝑥 ∈ {+∞, -∞})
111, 10mprgbir 2379 1 (ℝ ∩ {+∞, -∞}) = ∅
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∨ wo 629   = wceq 1243   ∈ wcel 1393   ∩ cin 2916  ∅c0 3224  {cpr 3376  ℝcr 6888  +∞cpnf 7057  -∞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-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-un 4170  ax-setind 4262  ax-cnex 6975  ax-resscn 6976 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  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-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-pnf 7062  df-mnf 7063 This theorem is referenced by: (None)
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