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Theorem iccgelbd 42546
 Description: An element of a closed interval is more than or equal to its lower bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
iccgelbd.1 (𝜑𝐴 ∈ ℝ*)
iccgelbd.2 (𝜑𝐵 ∈ ℝ*)
iccgelbd.3 (𝜑𝐶 ∈ (𝐴[,]𝐵))
Assertion
Ref Expression
iccgelbd (𝜑𝐴𝐶)

Proof of Theorem iccgelbd
StepHypRef Expression
1 iccgelbd.1 . 2 (𝜑𝐴 ∈ ℝ*)
2 iccgelbd.2 . 2 (𝜑𝐵 ∈ ℝ*)
3 iccgelbd.3 . 2 (𝜑𝐶 ∈ (𝐴[,]𝐵))
4 iccgelb 12835 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴[,]𝐵)) → 𝐴𝐶)
51, 2, 3, 4syl3anc 1368 1 (𝜑𝐴𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2111   class class class wbr 5032  (class class class)co 7150  ℝ*cxr 10712   ≤ cle 10714  [,]cicc 12782 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-iota 6294  df-fun 6337  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-xr 10717  df-icc 12786 This theorem is referenced by:  sqrlearg  42556
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