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Theorem iocgtlb 45405
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 13419 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ*𝐴 < 𝐶𝐶𝐵)))
2 simp2 1135 . . 3 ((𝐶 ∈ ℝ*𝐴 < 𝐶𝐶𝐵) → 𝐴 < 𝐶)
31, 2biimtrdi 253 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,]𝐵) → 𝐴 < 𝐶))
433impia 1115 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴(,]𝐵)) → 𝐴 < 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1085  wcel 2104   class class class wbr 5149  (class class class)co 7425  *cxr 11285   < clt 11286  cle 11287  (,]cioc 13378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5430  ax-un 7747  ax-cnex 11202  ax-resscn 11203
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2536  df-eu 2565  df-clab 2711  df-cleq 2725  df-clel 2812  df-nfc 2888  df-ral 3058  df-rex 3067  df-rab 3433  df-v 3479  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-br 5150  df-opab 5212  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-iota 6510  df-fun 6560  df-fv 6566  df-ov 7428  df-oprab 7429  df-mpo 7430  df-xr 11290  df-ioc 13382
This theorem is referenced by:  iocopn  45423  eliccelioc  45424  iccdificc  45442  iocgtlbd  45474  limcresiooub  45548  fourierdlem19  46032  fourierdlem35  46048  fourierdlem41  46054  fourierdlem46  46058  fourierdlem48  46060  fourierdlem49  46061  fourierdlem51  46063  fourierswlem  46136  fouriersw  46137
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