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Theorem elioo3g 13413
Description: Membership in a set of open intervals of extended reals. We use the fact that an operation's value is empty outside of its domain to show 𝐴 ∈ ℝ* and 𝐵 ∈ ℝ*. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
Assertion
Ref Expression
elioo3g (𝐶 ∈ (𝐴(,)𝐵) ↔ ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐶𝐶 < 𝐵)))

Proof of Theorem elioo3g
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ioo 13388 . 2 (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧𝑧 < 𝑦)})
21elixx3g 13397 1 (𝐶 ∈ (𝐴(,)𝐵) ↔ ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐶𝐶 < 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1086  wcel 2106   class class class wbr 5148  (class class class)co 7431  *cxr 11292   < clt 11293  (,)cioo 13384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-xr 11297  df-ioo 13388
This theorem is referenced by:  elioore  13414  lbioo  13415  ubioo  13416  elioo4g  13444  zltaddlt1le  13542  halfleoddlt  16396  qdensere  24806  cnndvlem1  36520  lptioo2  45587  lptioo1  45588  icccncfext  45843  iblcncfioo  45934  fourierdlem12  46075  fourierdlem74  46136  fourierdlem75  46137  fourierdlem103  46165  iccpartnel  47363
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