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Theorem iocssicc 12675
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 12593 . 2 (,] = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑥 ∈ ℝ* ∣ (𝑎 < 𝑥𝑥𝑏)})
2 df-icc 12595 . 2 [,] = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑥 ∈ ℝ* ∣ (𝑎𝑥𝑥𝑏)})
3 xrltle 12392 . 2 ((𝐴 ∈ ℝ*𝑤 ∈ ℝ*) → (𝐴 < 𝑤𝐴𝑤))
4 idd 24 . 2 ((𝑤 ∈ ℝ*𝐵 ∈ ℝ*) → (𝑤𝐵𝑤𝐵))
51, 2, 3, 4ixxssixx 12602 1 (𝐴(,]𝐵) ⊆ (𝐴[,]𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 396  wcel 2081  wss 3859   class class class wbr 4962  (class class class)co 7016  *cxr 10520   < clt 10521  cle 10522  (,]cioc 12589  [,]cicc 12591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-cnex 10439  ax-resscn 10440  ax-pre-lttri 10457  ax-pre-lttrn 10458
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-po 5362  df-so 5363  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-er 8139  df-en 8358  df-dom 8359  df-sdom 8360  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-ioc 12593  df-icc 12595
This theorem is referenced by:  xrge0iifcnv  30793  xrge0iifcv  30794  xrge0iifhom  30797  pnfneige0  30811  lmxrge0  30812  eliccelioc  41339  limcicciooub  41460  fourierdlem17  41951  fourierdlem35  41969  fourierdlem41  41975  fourierdlem48  41981  fourierdlem49  41982  fourierdlem51  41984  fourierdlem71  42004  fourierdlem102  42035  fourierdlem114  42047
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