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Theorem iccleub 12168
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 12158 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ*𝐴𝐶𝐶𝐵)))
2 simp3 1061 . . 3 ((𝐶 ∈ ℝ*𝐴𝐶𝐶𝐵) → 𝐶𝐵)
31, 2syl6bi 243 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) → 𝐶𝐵))
433impia 1258 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴[,]𝐵)) → 𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036  wcel 1992   class class class wbr 4618  (class class class)co 6605  *cxr 10018  cle 10020  [,]cicc 12117
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5813  df-fun 5852  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-xr 10023  df-icc 12121
This theorem is referenced by:  supicc  12259  supiccub  12260  supicclub  12261  oprpiece1res1  22653  ivthlem1  23122  isosctrlem1  24443  ttgcontlem1  25660  broucube  33042  mblfinlem1  33045  ftc1cnnclem  33082  ftc2nc  33093  areaquad  37250  isosctrlem1ALT  38620  lefldiveq  38937  eliccelioc  39126  iccintsng  39128  eliccnelico  39135  eliccelicod  39136  inficc  39140  iccdificc  39145  iccleubd  39154  cncfiooiccre  39380  itgioocnicc  39468  itgspltprt  39470  itgiccshift  39471  fourierdlem1  39600  fourierdlem20  39619  fourierdlem24  39623  fourierdlem25  39624  fourierdlem27  39626  fourierdlem43  39642  fourierdlem44  39643  fourierdlem50  39648  fourierdlem51  39649  fourierdlem52  39650  fourierdlem64  39662  fourierdlem73  39671  fourierdlem76  39674  fourierdlem79  39677  fourierdlem81  39679  fourierdlem92  39690  fourierdlem102  39700  fourierdlem103  39701  fourierdlem104  39702  fourierdlem114  39712  rrxsnicc  39795  salgencntex  39836  sge0p1  39906  hoidmv1lelem3  40082  hoidmvlelem1  40084  hoidmvlelem4  40087
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