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Theorem elicc1 12161
Description: Membership in a closed interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
Assertion
Ref Expression
elicc1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ*𝐴𝐶𝐶𝐵)))

Proof of Theorem elicc1
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-icc 12124 . 2 [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧𝑦)})
21elixx1 12126 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ*𝐴𝐶𝐶𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wcel 1987   class class class wbr 4613  (class class class)co 6604  *cxr 10017  cle 10019  [,]cicc 12120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-xr 10022  df-icc 12124
This theorem is referenced by:  iccid  12162  iccleub  12171  iccgelb  12172  elicc2  12180  elicc4  12182  xrge0neqmnf  12218  elxrge0  12223  lbicc2  12230  ubicc2  12231  difreicc  12246  cnblcld  22488  oprpiece1res1  22658  ovolf  23157  volivth  23281  itg2ge0  23408  itg2const2  23414  taylfvallem1  24015  tayl0  24020  radcnvcl  24075  radcnvle  24078  psercnlem1  24083  eliccelico  29380  xrdifh  29383  unitssxrge0  29725  esumle  29898  esumlef  29902  esumpinfsum  29917  voliune  30070  volfiniune  30071  ddemeas  30077  prob01  30253  elicc3  31950  ftc1cnnclem  33112  ftc1anc  33122  ftc2nc  33123  iocinico  37275  icoiccdif  39158  iblsplit  39486  iblspltprt  39493  itgspltprt  39499  fourierdlem1  39629  iccpartrn  40661
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