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Theorem iocssicc 9959
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 9891 . 2 (,] = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑥 ∈ ℝ* ∣ (𝑎 < 𝑥𝑥𝑏)})
2 df-icc 9893 . 2 [,] = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑥 ∈ ℝ* ∣ (𝑎𝑥𝑥𝑏)})
3 xrltle 9796 . 2 ((𝐴 ∈ ℝ*𝑤 ∈ ℝ*) → (𝐴 < 𝑤𝐴𝑤))
4 idd 21 . 2 ((𝑤 ∈ ℝ*𝐵 ∈ ℝ*) → (𝑤𝐵𝑤𝐵))
51, 2, 3, 4ixxssixx 9900 1 (𝐴(,]𝐵) ⊆ (𝐴[,]𝐵)
Colors of variables: wff set class
Syntax hints:  wa 104  wcel 2148  wss 3129   class class class wbr 4003  (class class class)co 5874  *cxr 7989   < clt 7990  cle 7991  (,]cioc 9887  [,]cicc 9889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-pre-ltirr 7922  ax-pre-lttrn 7924
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-ioc 9891  df-icc 9893
This theorem is referenced by: (None)
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