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Theorem icoltubd 44812
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 44775 . 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 2098   class class class wbr 5141  (class class class)co 7404  *cxr 11248   < clt 11249  [,)cico 13329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-resscn 11166
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-xr 11253  df-ico 13333
This theorem is referenced by:  icomnfinre  44819  xlimmnfvlem1  45102
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