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Theorem iccleub 13427
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 13416 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ*𝐴𝐶𝐶𝐵)))
2 simp3 1135 . . 3 ((𝐶 ∈ ℝ*𝐴𝐶𝐶𝐵) → 𝐶𝐵)
31, 2biimtrdi 252 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) → 𝐶𝐵))
433impia 1114 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴[,]𝐵)) → 𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084  wcel 2099   class class class wbr 5145  (class class class)co 7416  *cxr 11288  cle 11290  [,]cicc 13375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425  ax-un 7738  ax-cnex 11205  ax-resscn 11206
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3776  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-iota 6498  df-fun 6548  df-fv 6554  df-ov 7419  df-oprab 7420  df-mpo 7421  df-xr 11293  df-icc 13379
This theorem is referenced by:  supicc  13526  supiccub  13527  supicclub  13528  oprpiece1res1  24964  ivthlem1  25468  isosctrlem1  26843  ttgcontlem1  28815  broucube  37368  mblfinlem1  37371  ftc1cnnclem  37405  ftc2nc  37416  areaquad  42918  isosctrlem1ALT  44647  lefldiveq  44943  eliccelioc  45175  iccintsng  45177  eliccnelico  45183  eliccelicod  45184  inficc  45188  iccdificc  45193  iccleubd  45202  cncfiooiccre  45552  itgioocnicc  45634  itgspltprt  45636  itgiccshift  45637  fourierdlem1  45765  fourierdlem20  45784  fourierdlem24  45788  fourierdlem25  45789  fourierdlem27  45791  fourierdlem43  45807  fourierdlem44  45808  fourierdlem50  45813  fourierdlem51  45814  fourierdlem52  45815  fourierdlem64  45827  fourierdlem73  45836  fourierdlem76  45839  fourierdlem79  45842  fourierdlem81  45844  fourierdlem92  45855  fourierdlem102  45865  fourierdlem103  45866  fourierdlem104  45867  fourierdlem114  45877  rrxsnicc  45957  salgencntex  46000  sge0p1  46071  hoidmv1lelem3  46250  hoidmvlelem1  46252  hoidmvlelem4  46255
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