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Theorem icogelbd 39588
Description: An element of a left closed right open interval is larger or equal to its lower bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
icogelbd.1 (𝜑𝐴 ∈ ℝ*)
icogelbd.2 (𝜑𝐵 ∈ ℝ*)
icogelbd.3 (𝜑𝐶 ∈ (𝐴[,)𝐵))
Assertion
Ref Expression
icogelbd (𝜑𝐴𝐶)

Proof of Theorem icogelbd
StepHypRef Expression
1 icogelbd.1 . 2 (𝜑𝐴 ∈ ℝ*)
2 icogelbd.2 . 2 (𝜑𝐵 ∈ ℝ*)
3 icogelbd.3 . 2 (𝜑𝐶 ∈ (𝐴[,)𝐵))
4 icogelb 12210 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴[,)𝐵)) → 𝐴𝐶)
51, 2, 3, 4syl3anc 1324 1 (𝜑𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1988   class class class wbr 4644  (class class class)co 6635  *cxr 10058  cle 10060  [,)cico 12162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-xr 10063  df-ico 12166
This theorem is referenced by:  uzinico  39590  limsupresico  39732  limsupmnflem  39752  liminfresico  39797  liminflelimsuplem  39801  smfliminflem  40799
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