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Theorem elicc1 12424
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 12387 . 2 [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧𝑦)})
21elixx1 12389 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ*𝐴𝐶𝐶𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071  wcel 2145   class class class wbr 4786  (class class class)co 6793  *cxr 10275  cle 10277  [,]cicc 12383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-xr 10280  df-icc 12387
This theorem is referenced by:  iccid  12425  iccleub  12434  iccgelb  12435  elicc2  12443  elicc4  12445  xrge0neqmnfOLD  12483  elxrge0  12488  lbicc2  12495  ubicc2  12496  difreicc  12511  cnblcld  22798  oprpiece1res1  22970  ovolf  23470  volivth  23595  itg2ge0  23722  itg2const2  23728  taylfvallem1  24331  tayl0  24336  radcnvcl  24391  radcnvle  24394  psercnlem1  24399  eliccelico  29879  xrdifh  29882  unitssxrge0  30286  esumle  30460  esumlef  30464  esumpinfsum  30479  voliune  30632  volfiniune  30633  ddemeas  30639  prob01  30815  elicc3  32648  ftc1cnnclem  33815  ftc1anc  33825  ftc2nc  33826  iocinico  38323  icoiccdif  40269  iblsplit  40699  iblspltprt  40706  itgspltprt  40712  fourierdlem1  40842  iccpartrn  41894
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