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Theorem icoltub 40235
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 12411 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ*𝐴𝐶𝐶 < 𝐵)))
2 simp3 1133 . . 3 ((𝐶 ∈ ℝ*𝐴𝐶𝐶 < 𝐵) → 𝐶 < 𝐵)
31, 2syl6bi 243 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,)𝐵) → 𝐶 < 𝐵))
433impia 1110 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 < 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072  wcel 2139   class class class wbr 4804  (class class class)co 6813  *cxr 10265   < clt 10266  cle 10267  [,)cico 12370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-xr 10270  df-ico 12374
This theorem is referenced by:  icoopn  40254  icoub  40255  icoltubd  40275  ltmod  40373  limcresioolb  40378  fourierdlem41  40868  fourierdlem43  40870  fourierdlem46  40872  fourierdlem48  40874  fouriersw  40951  hoidmv1lelem2  41312  hoidmvlelem2  41316  hspdifhsp  41336  hspmbllem2  41347  iinhoiicclem  41393  preimaicomnf  41428
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