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Theorem icossxr 12819
Description: A closed-below, open-above interval is a subset of the extended reals. (Contributed by FL, 29-May-2014.) (Revised by Mario Carneiro, 4-Jul-2014.)
Assertion
Ref Expression
icossxr (𝐴[,)𝐵) ⊆ ℝ*

Proof of Theorem icossxr
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ico 12741 . 2 [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
21ixxssxr 12747 1 (𝐴[,)𝐵) ⊆ ℝ*
Colors of variables: wff setvar class
Syntax hints:  wss 3919  (class class class)co 7149  *cxr 10672   < clt 10673  cle 10674  [,)cico 12737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685  df-xr 10677  df-ico 12741
This theorem is referenced by:  leordtvallem2  21823  leordtval2  21824  nmoffn  23324  nmofval  23327  nmogelb  23329  nmolb  23330  nmof  23332  icopnfhmeo  23555  elovolm  24086  ovolmge0  24088  ovolgelb  24091  ovollb2lem  24099  ovoliunlem1  24113  ovoliunlem2  24114  ovolscalem1  24124  ovolicc1  24127  ioombl1lem2  24170  ioombl1lem4  24172  uniioovol  24190  uniiccvol  24191  uniioombllem1  24192  uniioombllem2  24194  uniioombllem3  24196  uniioombllem6  24199  esumpfinvallem  31394  esummulc1  31401  esummulc2  31402  mblfinlem3  35045  mblfinlem4  35046  ismblfin  35047  itg2gt0cn  35061  xralrple2  41917  icoub  42094  liminflelimsuplem  42348  elhoi  43112  hoidmvlelem5  43169  ovnhoilem1  43171  ovnhoilem2  43172  ovnhoi  43173
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