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Theorem icoltubd 42180
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 42143 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 < 𝐵)
51, 2, 3, 4syl3anc 1368 1 (𝜑𝐶 < 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2111   class class class wbr 5030  (class class class)co 7135  *cxr 10663   < clt 10664  [,)cico 12728
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-xr 10668  df-ico 12732
This theorem is referenced by:  icomnfinre  42187  xlimmnfvlem1  42472
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