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Theorem icoltub 38579
Description: An element of a left closed right open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Assertion
Ref Expression
icoltub ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 < 𝐵)

Proof of Theorem icoltub
StepHypRef Expression
1 elico1 12089 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ*𝐴𝐶𝐶 < 𝐵)))
2 simp3 1056 . . 3 ((𝐶 ∈ ℝ*𝐴𝐶𝐶 < 𝐵) → 𝐶 < 𝐵)
31, 2syl6bi 242 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,)𝐵) → 𝐶 < 𝐵))
433impia 1253 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 < 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031  wcel 1977   class class class wbr 4583  (class class class)co 6549  *cxr 9952   < clt 9953  cle 9954  [,)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:  icoopn  38598  icoub  38599  icoltubd  38619  ltmod  38705  limcresioolb  38710  fourierdlem41  39041  fourierdlem43  39043  fourierdlem46  39045  fourierdlem48  39047  fouriersw  39124  hoidmv1lelem2  39482  hoidmvlelem2  39486  hspdifhsp  39506  hspmbllem2  39517  iinhoiicclem  39564  preimaicomnf  39599
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