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Theorem iocleub 41931
Description: An element of a left-open right-closed interval is smaller than or equal to its upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Assertion
Ref Expression
iocleub ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴(,]𝐵)) → 𝐶𝐵)

Proof of Theorem iocleub
StepHypRef Expression
1 elioc1 12759 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ*𝐴 < 𝐶𝐶𝐵)))
2 simp3 1134 . . 3 ((𝐶 ∈ ℝ*𝐴 < 𝐶𝐶𝐵) → 𝐶𝐵)
31, 2syl6bi 255 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,]𝐵) → 𝐶𝐵))
433impia 1113 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴(,]𝐵)) → 𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1083  wcel 2114   class class class wbr 5042  (class class class)co 7133  *cxr 10652   < clt 10653  cle 10654  (,]cioc 12718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306  ax-un 7439  ax-cnex 10571  ax-resscn 10572
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-iota 6290  df-fun 6333  df-fv 6339  df-ov 7136  df-oprab 7137  df-mpo 7138  df-xr 10657  df-ioc 12722
This theorem is referenced by:  iocopn  41948  iccdificc  41967  iocleubd  41987  limcresiooub  42075  fourierdlem19  42559  fourierdlem35  42575  fourierdlem41  42581  fourierdlem46  42585  fourierdlem48  42587  fourierdlem49  42588  fourierswlem  42663  fouriersw  42664  pimdecfgtioc  43141  pimincfltioc  43142
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