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Theorem icoltub 40467
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 12463 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ*𝐴𝐶𝐶 < 𝐵)))
2 simp3 1169 . . 3 ((𝐶 ∈ ℝ*𝐴𝐶𝐶 < 𝐵) → 𝐶 < 𝐵)
31, 2syl6bi 245 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,)𝐵) → 𝐶 < 𝐵))
433impia 1146 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 < 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108  wcel 2157   class class class wbr 4841  (class class class)co 6876  *cxr 10360   < clt 10361  cle 10362  [,)cico 12422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095  ax-un 7181  ax-cnex 10278  ax-resscn 10279
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-iota 6062  df-fun 6101  df-fv 6107  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-xr 10365  df-ico 12426
This theorem is referenced by:  icoopn  40484  icoub  40485  icoltubd  40504  ltmod  40602  limcresioolb  40607  fourierdlem41  41096  fourierdlem43  41098  fourierdlem46  41100  fourierdlem48  41102  fouriersw  41179  hoidmv1lelem2  41540  hoidmvlelem2  41544  hspdifhsp  41564  hspmbllem2  41575  iinhoiicclem  41621  preimaicomnf  41656
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