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Theorem iocssicc 12471
Description: A closed-above, open-below interval is a subset of its closure. (Contributed by Thierry Arnoux, 1-Apr-2017.)
Assertion
Ref Expression
iocssicc (𝐴(,]𝐵) ⊆ (𝐴[,]𝐵)

Proof of Theorem iocssicc
Dummy variables 𝑎 𝑏 𝑤 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ioc 12389 . 2 (,] = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑥 ∈ ℝ* ∣ (𝑎 < 𝑥𝑥𝑏)})
2 df-icc 12391 . 2 [,] = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑥 ∈ ℝ* ∣ (𝑎𝑥𝑥𝑏)})
3 xrltle 12189 . 2 ((𝐴 ∈ ℝ*𝑤 ∈ ℝ*) → (𝐴 < 𝑤𝐴𝑤))
4 idd 24 . 2 ((𝑤 ∈ ℝ*𝐵 ∈ ℝ*) → (𝑤𝐵𝑤𝐵))
51, 2, 3, 4ixxssixx 12398 1 (𝐴(,]𝐵) ⊆ (𝐴[,]𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 384  wcel 2155  wss 3734   class class class wbr 4811  (class class class)co 6846  *cxr 10331   < clt 10332  cle 10333  (,]cioc 12385  [,]cicc 12387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-cnex 10249  ax-resscn 10250  ax-pre-lttri 10267  ax-pre-lttrn 10268
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-po 5200  df-so 5201  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-er 7951  df-en 8165  df-dom 8166  df-sdom 8167  df-pnf 10334  df-mnf 10335  df-xr 10336  df-ltxr 10337  df-le 10338  df-ioc 12389  df-icc 12391
This theorem is referenced by:  xrge0iifcnv  30449  xrge0iifcv  30450  xrge0iifhom  30453  pnfneige0  30467  lmxrge0  30468  eliccelioc  40412  limcicciooub  40533  fourierdlem17  41004  fourierdlem35  41022  fourierdlem41  41028  fourierdlem48  41034  fourierdlem49  41035  fourierdlem51  41037  fourierdlem71  41057  fourierdlem102  41088  fourierdlem114  41100
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