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Theorem elicc1 12774
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 12737 . 2 [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧𝑦)})
21elixx1 12739 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ*𝐴𝐶𝐶𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084  wcel 2112   class class class wbr 5033  (class class class)co 7139  *cxr 10667  cle 10669  [,]cicc 12733
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587
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 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-xr 10672  df-icc 12737
This theorem is referenced by:  iccid  12775  iccleub  12784  iccgelb  12785  elicc2  12794  elicc4  12796  elxrge0  12839  lbicc2  12846  ubicc2  12847  difreicc  12866  cnblcld  23384  ovolf  24090  volivth  24215  itg2ge0  24343  itg2const2  24349  taylfvallem1  24956  tayl0  24961  radcnvcl  25016  radcnvle  25019  psercnlem1  25024  eliccelico  30530  xrdifh  30533  unitssxrge0  31257  esumle  31431  esumlef  31435  esumpinfsum  31450  voliune  31602  volfiniune  31603  ddemeas  31609  prob01  31785  elicc3  33779  ftc1cnnclem  35127  ftc1anc  35137  ftc2nc  35138  iocinico  40159  icoiccdif  42158  iblsplit  42605  iblspltprt  42612  itgspltprt  42618  fourierdlem1  42747  iccpartrn  43944  rrxsphere  45159
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