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Theorem iccgelbd 40660
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 12542 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴[,]𝐵)) → 𝐴𝐶)
51, 2, 3, 4syl3anc 1439 1 (𝜑𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106   class class class wbr 4886  (class class class)co 6922  *cxr 10410  cle 10412  [,]cicc 12490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-xr 10415  df-icc 12494
This theorem is referenced by:  sqrlearg  40670
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