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Theorem iocleubd 45512
Description: An element of a left-open right-closed interval is smaller than or equal to its upper bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
iocleubd.1 (𝜑𝐴 ∈ ℝ*)
iocleubd.2 (𝜑𝐵 ∈ ℝ*)
iocleubd.3 (𝜑𝐶 ∈ (𝐴(,]𝐵))
Assertion
Ref Expression
iocleubd (𝜑𝐶𝐵)

Proof of Theorem iocleubd
StepHypRef Expression
1 iocleubd.1 . 2 (𝜑𝐴 ∈ ℝ*)
2 iocleubd.2 . 2 (𝜑𝐵 ∈ ℝ*)
3 iocleubd.3 . 2 (𝜑𝐶 ∈ (𝐴(,]𝐵))
4 iocleub 45456 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴(,]𝐵)) → 𝐶𝐵)
51, 2, 3, 4syl3anc 1370 1 (𝜑𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106   class class class wbr 5148  (class class class)co 7431  *cxr 11292  cle 11294  (,]cioc 13385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-xr 11297  df-ioc 13389
This theorem is referenced by:  preimaiocmnf  45514  smfsuplem1  46767
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