Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  icoltub Structured version   Visualization version   GIF version

Theorem icoltub 39140
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 12160 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ*𝐴𝐶𝐶 < 𝐵)))
2 simp3 1061 . . 3 ((𝐶 ∈ ℝ*𝐴𝐶𝐶 < 𝐵) → 𝐶 < 𝐵)
31, 2syl6bi 243 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,)𝐵) → 𝐶 < 𝐵))
433impia 1258 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 < 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036  wcel 1987   class class class wbr 4613  (class class class)co 6604  *cxr 10017   < clt 10018  cle 10019  [,)cico 12119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-xr 10022  df-ico 12123
This theorem is referenced by:  icoopn  39159  icoub  39160  icoltubd  39180  ltmod  39271  limcresioolb  39276  fourierdlem41  39669  fourierdlem43  39671  fourierdlem46  39673  fourierdlem48  39675  fouriersw  39752  hoidmv1lelem2  40110  hoidmvlelem2  40114  hspdifhsp  40134  hspmbllem2  40145  iinhoiicclem  40191  preimaicomnf  40226
  Copyright terms: Public domain W3C validator