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Theorem iccgelb 12781
Description: An element of a closed interval is more than or equal to its lower bound. (Contributed by Thierry Arnoux, 23-Dec-2016.)
Assertion
Ref Expression
iccgelb ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴[,]𝐵)) → 𝐴𝐶)

Proof of Theorem iccgelb
StepHypRef Expression
1 elicc1 12770 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ*𝐴𝐶𝐶𝐵)))
21biimpa 477 . . 3 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ 𝐶 ∈ (𝐴[,]𝐵)) → (𝐶 ∈ ℝ*𝐴𝐶𝐶𝐵))
32simp2d 1135 . 2 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴𝐶)
433impa 1102 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴[,]𝐵)) → 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079  wcel 2105   class class class wbr 5057  (class class class)co 7145  *cxr 10662  cle 10664  [,]cicc 12729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-xr 10667  df-icc 12733
This theorem is referenced by:  xrge0neqmnf  12828  supicc  12874  ttgcontlem1  26598  xrge0infss  30410  xrge0addgt0  30605  xrge0adddir  30606  esumcst  31221  esumpinfval  31231  oms0  31454  probmeasb  31587  broucube  34807  areaquad  39701  lefldiveq  41435  xadd0ge  41464  xrge0nemnfd  41476  eliccelioc  41673  iccintsng  41675  eliccnelico  41681  eliccelicod  41682  ge0xrre  41683  inficc  41686  iccdificc  41691  iccgelbd  41695  cncfiooiccre  42054  iblspltprt  42134  itgioocnicc  42138  itgspltprt  42140  itgiccshift  42141  fourierdlem1  42270  fourierdlem20  42289  fourierdlem24  42293  fourierdlem25  42294  fourierdlem27  42296  fourierdlem43  42312  fourierdlem44  42313  fourierdlem50  42318  fourierdlem51  42319  fourierdlem52  42320  fourierdlem64  42332  fourierdlem73  42341  fourierdlem76  42344  fourierdlem81  42349  fourierdlem92  42360  fourierdlem102  42370  fourierdlem103  42371  fourierdlem104  42372  fourierdlem114  42382  rrxsnicc  42462  salgencntex  42503  fge0iccico  42529  gsumge0cl  42530  sge0sn  42538  sge0tsms  42539  sge0cl  42540  sge0ge0  42543  sge0fsum  42546  sge0pr  42553  sge0prle  42560  sge0p1  42573  sge0rernmpt  42581  meage0  42634  omessre  42669  omeiunltfirp  42678  carageniuncllem2  42681  omege0  42692  ovnlerp  42721  ovn0lem  42724  hoidmvlelem1  42754  hoidmvlelem2  42755  hoidmvlelem3  42756
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