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Theorem iocgtlb 40523
 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 12505 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ*𝐴 < 𝐶𝐶𝐵)))
2 simp2 1173 . . 3 ((𝐶 ∈ ℝ*𝐴 < 𝐶𝐶𝐵) → 𝐴 < 𝐶)
31, 2syl6bi 245 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,]𝐵) → 𝐴 < 𝐶))
433impia 1151 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴(,]𝐵)) → 𝐴 < 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   ∧ w3a 1113   ∈ wcel 2166   class class class wbr 4873  (class class class)co 6905  ℝ*cxr 10390   < clt 10391   ≤ cle 10392  (,]cioc 12464 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127  ax-un 7209  ax-cnex 10308  ax-resscn 10309 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-iota 6086  df-fun 6125  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-xr 10395  df-ioc 12468 This theorem is referenced by:  iocopn  40542  eliccelioc  40543  iccdificc  40561  iocgtlbd  40593  limcresiooub  40669  fourierdlem19  41137  fourierdlem35  41153  fourierdlem41  41159  fourierdlem46  41163  fourierdlem48  41165  fourierdlem49  41166  fourierdlem51  41168  fourierswlem  41241  fouriersw  41242
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