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Theorem iccleubd 44098
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 13363 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴[,]𝐵)) → 𝐶𝐵)
51, 2, 3, 4syl3anc 1371 1 (𝜑𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106   class class class wbr 5142  (class class class)co 7394  *cxr 11231  cle 11233  [,]cicc 13311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5293  ax-nul 5300  ax-pr 5421  ax-un 7709  ax-cnex 11150  ax-resscn 11151
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3775  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5143  df-opab 5205  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-iota 6485  df-fun 6535  df-fv 6541  df-ov 7397  df-oprab 7398  df-mpo 7399  df-xr 11236  df-icc 13315
This theorem is referenced by:  sqrlearg  44103
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