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Theorem iccgelbd 40088
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 12268 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴[,]𝐵)) → 𝐴𝐶)
51, 2, 3, 4syl3anc 1366 1 (𝜑𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2030   class class class wbr 4685  (class class class)co 6690  *cxr 10111  cle 10113  [,]cicc 12216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-xr 10116  df-icc 12220
This theorem is referenced by:  sqrlearg  40098
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