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Theorem icoltubd 44193
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 44156 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 < 𝐵)
51, 2, 3, 4syl3anc 1372 1 (𝜑𝐶 < 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107   class class class wbr 5147  (class class class)co 7404  *cxr 11243   < clt 11244  [,)cico 13322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7720  ax-cnex 11162  ax-resscn 11163
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7407  df-oprab 7408  df-mpo 7409  df-xr 11248  df-ico 13326
This theorem is referenced by:  icomnfinre  44200  xlimmnfvlem1  44483
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