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Theorem eliooxr 9026
 Description: An inhabited open interval spans an interval of extended reals. (Contributed by NM, 17-Aug-2008.)
Assertion
Ref Expression
eliooxr (𝐴 ∈ (𝐵(,)𝐶) → (𝐵 ∈ ℝ*𝐶 ∈ ℝ*))

Proof of Theorem eliooxr
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ioo 8991 . 2 (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧𝑧 < 𝑦)})
21elmpt2cl 5729 1 (𝐴 ∈ (𝐵(,)𝐶) → (𝐵 ∈ ℝ*𝐶 ∈ ℝ*))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102   ∈ wcel 1434  {crab 2353   class class class wbr 3793  (class class class)co 5543  ℝ*cxr 7214   < clt 7215  (,)cioo 8987 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-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-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-ioo 8991 This theorem is referenced by:  eliooord  9027  elioo4g  9033  iccssioo2  9045
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