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Theorem iccleubd 45500
Description: An element of a closed interval is less than or equal to its upper bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
iccleubd.1 (𝜑𝐴 ∈ ℝ*)
iccleubd.2 (𝜑𝐵 ∈ ℝ*)
iccleubd.3 (𝜑𝐶 ∈ (𝐴[,]𝐵))
Assertion
Ref Expression
iccleubd (𝜑𝐶𝐵)

Proof of Theorem iccleubd
StepHypRef Expression
1 iccleubd.1 . 2 (𝜑𝐴 ∈ ℝ*)
2 iccleubd.2 . 2 (𝜑𝐵 ∈ ℝ*)
3 iccleubd.3 . 2 (𝜑𝐶 ∈ (𝐴[,]𝐵))
4 iccleub 13438 . 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 2105   class class class wbr 5147  (class class class)co 7430  *cxr 11291  cle 11293  [,]cicc 13386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-xr 11296  df-icc 13390
This theorem is referenced by:  sqrlearg  45505
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