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Theorem iocgtlb 42069
Description: An element of a left-open right-closed interval is larger than its lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Assertion
Ref Expression
iocgtlb ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴(,]𝐵)) → 𝐴 < 𝐶)

Proof of Theorem iocgtlb
StepHypRef Expression
1 elioc1 12777 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ*𝐴 < 𝐶𝐶𝐵)))
2 simp2 1134 . . 3 ((𝐶 ∈ ℝ*𝐴 < 𝐶𝐶𝐵) → 𝐴 < 𝐶)
31, 2syl6bi 256 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,]𝐵) → 𝐴 < 𝐶))
433impia 1114 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴(,]𝐵)) → 𝐴 < 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084  wcel 2115   class class class wbr 5052  (class class class)co 7149  *cxr 10672   < clt 10673  cle 10674  (,]cioc 12736
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-iota 6302  df-fun 6345  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-xr 10677  df-ioc 12740
This theorem is referenced by:  iocopn  42087  eliccelioc  42088  iccdificc  42106  iocgtlbd  42138  limcresiooub  42214  fourierdlem19  42698  fourierdlem35  42714  fourierdlem41  42720  fourierdlem46  42724  fourierdlem48  42726  fourierdlem49  42727  fourierdlem51  42729  fourierswlem  42802  fouriersw  42803
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