Theorem icoltubd 46152

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	⊢ (𝜑 → 𝐴 ∈ ℝ*)
icoltubd.2	⊢ (𝜑 → 𝐵 ∈ ℝ*)
icoltubd.3	⊢ (𝜑 → 𝐶 ∈ (𝐴[,)𝐵))

Assertion

Ref	Expression
icoltubd	⊢ (𝜑 → 𝐶 < 𝐵)

Proof of Theorem icoltubd

Step	Hyp	Ref	Expression
1		icoltubd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ*)
2		icoltubd.2	. 2 ⊢ (𝜑 → 𝐵 ∈ ℝ*)
3		icoltubd.3	. 2 ⊢ (𝜑 → 𝐶 ∈ (𝐴[,)𝐵))
4		icoltub 46115	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 < 𝐵)
5	1, 2, 3, 4	syl3anc 1396	1 ⊢ (𝜑 → 𝐶 < 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2149   class class class wbr 5113  (class class class)co 7411  ℝ^*cxr 11241   < clt 11242  [,)cico 13373

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-pr 5405  ax-un 7733  ax-cnex 11155  ax-resscn 11156

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-xr 11246  df-ico 13377

This theorem is referenced by:  icomnfinre  46159  xlimmnfvlem1  46437  fourierdlem48  46759