ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  iccss GIF version

Theorem iccss 9973
Description: Condition for a closed interval to be a subset of another closed interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 20-Feb-2015.)
Assertion
Ref Expression
iccss (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴𝐶𝐷𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,]𝐵))

Proof of Theorem iccss
Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexr 8034 . . 3 (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*)
2 rexr 8034 . . 3 (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*)
31, 2anim12i 338 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ∈ ℝ*𝐵 ∈ ℝ*))
4 df-icc 9927 . . 3 [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧𝑦)})
5 xrletr 9840 . . 3 ((𝐴 ∈ ℝ*𝐶 ∈ ℝ*𝑤 ∈ ℝ*) → ((𝐴𝐶𝐶𝑤) → 𝐴𝑤))
6 xrletr 9840 . . 3 ((𝑤 ∈ ℝ*𝐷 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝑤𝐷𝐷𝐵) → 𝑤𝐵))
74, 4, 5, 6ixxss12 9938 . 2 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐴𝐶𝐷𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,]𝐵))
83, 7sylan 283 1 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴𝐶𝐷𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,]𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wcel 2160  wss 3144   class class class wbr 4018  (class class class)co 5897  cr 7841  *cxr 8022  cle 8024  [,]cicc 9923
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-po 4314  df-iso 4315  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-icc 9927
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator