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Theorem icoltubd 38619
 Description: An element of a left closed right open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
icoltubd.1 (𝜑𝐴 ∈ ℝ*)
icoltubd.2 (𝜑𝐵 ∈ ℝ*)
icoltubd.3 (𝜑𝐶 ∈ (𝐴[,)𝐵))
Assertion
Ref Expression
icoltubd (𝜑𝐶 < 𝐵)

Proof of Theorem icoltubd
StepHypRef Expression
1 icoltubd.1 . 2 (𝜑𝐴 ∈ ℝ*)
2 icoltubd.2 . 2 (𝜑𝐵 ∈ ℝ*)
3 icoltubd.3 . 2 (𝜑𝐶 ∈ (𝐴[,)𝐵))
4 icoltub 38579 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 < 𝐵)
51, 2, 3, 4syl3anc 1318 1 (𝜑𝐶 < 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977   class class class wbr 4583  (class class class)co 6549  ℝ*cxr 9952   < clt 9953  [,)cico 12048 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-xr 9957  df-ico 12052 This theorem is referenced by:  icomnfinre  38626
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