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Theorem iocleubd 40304
Description: An element of a left open right closed interval is smaller or equal to its upper bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
iocleubd.1 (𝜑𝐴 ∈ ℝ*)
iocleubd.2 (𝜑𝐵 ∈ ℝ*)
iocleubd.3 (𝜑𝐶 ∈ (𝐴(,]𝐵))
Assertion
Ref Expression
iocleubd (𝜑𝐶𝐵)

Proof of Theorem iocleubd
StepHypRef Expression
1 iocleubd.1 . 2 (𝜑𝐴 ∈ ℝ*)
2 iocleubd.2 . 2 (𝜑𝐵 ∈ ℝ*)
3 iocleubd.3 . 2 (𝜑𝐶 ∈ (𝐴(,]𝐵))
4 iocleub 40246 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴(,]𝐵)) → 𝐶𝐵)
51, 2, 3, 4syl3anc 1476 1 (𝜑𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2145   class class class wbr 4786  (class class class)co 6793  *cxr 10275  cle 10277  (,]cioc 12381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-xr 10280  df-ioc 12385
This theorem is referenced by:  preimaiocmnf  40306  smfsuplem1  41537
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