MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iocssicc Structured version   Visualization version   GIF version

Theorem iocssicc 12246
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 12165 . 2 (,] = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑥 ∈ ℝ* ∣ (𝑎 < 𝑥𝑥𝑏)})
2 df-icc 12167 . 2 [,] = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑥 ∈ ℝ* ∣ (𝑎𝑥𝑥𝑏)})
3 xrltle 11967 . 2 ((𝐴 ∈ ℝ*𝑤 ∈ ℝ*) → (𝐴 < 𝑤𝐴𝑤))
4 idd 24 . 2 ((𝑤 ∈ ℝ*𝐵 ∈ ℝ*) → (𝑤𝐵𝑤𝐵))
51, 2, 3, 4ixxssixx 12174 1 (𝐴(,]𝐵) ⊆ (𝐴[,]𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 384  wcel 1988  wss 3567   class class class wbr 4644  (class class class)co 6635  *cxr 10058   < clt 10059  cle 10060  (,]cioc 12161  [,]cicc 12163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-pre-lttri 9995  ax-pre-lttrn 9996
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-po 5025  df-so 5026  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-ioc 12165  df-icc 12167
This theorem is referenced by:  xrge0iifcnv  29953  xrge0iifcv  29954  xrge0iifhom  29957  pnfneige0  29971  lmxrge0  29972  eliccelioc  39550  limcicciooub  39669  fourierdlem17  40104  fourierdlem35  40122  fourierdlem41  40128  fourierdlem48  40134  fourierdlem49  40135  fourierdlem51  40137  fourierdlem71  40157  fourierdlem102  40188  fourierdlem114  40200
  Copyright terms: Public domain W3C validator