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Theorem iccleub 12267
Description: An element of a closed interval is less than or equal to its upper bound. (Contributed by Jeff Hankins, 14-Jul-2009.)
Assertion
Ref Expression
iccleub ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴[,]𝐵)) → 𝐶𝐵)

Proof of Theorem iccleub
StepHypRef Expression
1 elicc1 12257 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ*𝐴𝐶𝐶𝐵)))
2 simp3 1083 . . 3 ((𝐶 ∈ ℝ*𝐴𝐶𝐶𝐵) → 𝐶𝐵)
31, 2syl6bi 243 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) → 𝐶𝐵))
433impia 1280 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴[,]𝐵)) → 𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054  wcel 2030   class class class wbr 4685  (class class class)co 6690  *cxr 10111  cle 10113  [,]cicc 12216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-xr 10116  df-icc 12220
This theorem is referenced by:  supicc  12358  supiccub  12359  supicclub  12360  oprpiece1res1  22797  ivthlem1  23266  isosctrlem1  24593  ttgcontlem1  25810  broucube  33573  mblfinlem1  33576  ftc1cnnclem  33613  ftc2nc  33624  areaquad  38119  isosctrlem1ALT  39484  lefldiveq  39819  eliccelioc  40065  iccintsng  40067  eliccnelico  40074  eliccelicod  40075  inficc  40079  iccdificc  40084  iccleubd  40093  cncfiooiccre  40426  itgioocnicc  40511  itgspltprt  40513  itgiccshift  40514  fourierdlem1  40643  fourierdlem20  40662  fourierdlem24  40666  fourierdlem25  40667  fourierdlem27  40669  fourierdlem43  40685  fourierdlem44  40686  fourierdlem50  40691  fourierdlem51  40692  fourierdlem52  40693  fourierdlem64  40705  fourierdlem73  40714  fourierdlem76  40717  fourierdlem79  40720  fourierdlem81  40722  fourierdlem92  40733  fourierdlem102  40743  fourierdlem103  40744  fourierdlem104  40745  fourierdlem114  40755  rrxsnicc  40838  salgencntex  40879  sge0p1  40949  hoidmv1lelem3  41128  hoidmvlelem1  41130  hoidmvlelem4  41133
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