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Theorem iocgtlb 44768
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 13369 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ*𝐴 < 𝐶𝐶𝐵)))
2 simp2 1134 . . 3 ((𝐶 ∈ ℝ*𝐴 < 𝐶𝐶𝐵) → 𝐴 < 𝐶)
31, 2syl6bi 253 . 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 395  w3a 1084  wcel 2098   class class class wbr 5141  (class class class)co 7404  *cxr 11248   < clt 11249  cle 11250  (,]cioc 13328
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-ioc 13332
This theorem is referenced by:  iocopn  44786  eliccelioc  44787  iccdificc  44805  iocgtlbd  44837  limcresiooub  44911  fourierdlem19  45395  fourierdlem35  45411  fourierdlem41  45417  fourierdlem46  45421  fourierdlem48  45423  fourierdlem49  45424  fourierdlem51  45426  fourierswlem  45499  fouriersw  45500
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