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Theorem iccss2 13455
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, 28-Apr-2015.)
Assertion
Ref Expression
iccss2 ((𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐷 ∈ (𝐴[,]𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,]𝐵))

Proof of Theorem iccss2
Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-icc 13391 . . . . . 6 [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧𝑦)})
21elixx3g 13397 . . . . 5 (𝐶 ∈ (𝐴[,]𝐵) ↔ ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐴𝐶𝐶𝐵)))
32simplbi 497 . . . 4 (𝐶 ∈ (𝐴[,]𝐵) → (𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*))
43adantr 480 . . 3 ((𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐷 ∈ (𝐴[,]𝐵)) → (𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*))
54simp1d 1141 . 2 ((𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐷 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ*)
64simp2d 1142 . 2 ((𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐷 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ*)
72simprbi 496 . . . 4 (𝐶 ∈ (𝐴[,]𝐵) → (𝐴𝐶𝐶𝐵))
87adantr 480 . . 3 ((𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐷 ∈ (𝐴[,]𝐵)) → (𝐴𝐶𝐶𝐵))
98simpld 494 . 2 ((𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐷 ∈ (𝐴[,]𝐵)) → 𝐴𝐶)
101elixx3g 13397 . . . . 5 (𝐷 ∈ (𝐴[,]𝐵) ↔ ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐷 ∈ ℝ*) ∧ (𝐴𝐷𝐷𝐵)))
1110simprbi 496 . . . 4 (𝐷 ∈ (𝐴[,]𝐵) → (𝐴𝐷𝐷𝐵))
1211simprd 495 . . 3 (𝐷 ∈ (𝐴[,]𝐵) → 𝐷𝐵)
1312adantl 481 . 2 ((𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐷 ∈ (𝐴[,]𝐵)) → 𝐷𝐵)
14 xrletr 13197 . . 3 ((𝐴 ∈ ℝ*𝐶 ∈ ℝ*𝑤 ∈ ℝ*) → ((𝐴𝐶𝐶𝑤) → 𝐴𝑤))
15 xrletr 13197 . . 3 ((𝑤 ∈ ℝ*𝐷 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝑤𝐷𝐷𝐵) → 𝑤𝐵))
161, 1, 14, 15ixxss12 13404 . 2 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐴𝐶𝐷𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,]𝐵))
175, 6, 9, 13, 16syl22anc 839 1 ((𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐷 ∈ (𝐴[,]𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,]𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086  wcel 2106  wss 3963   class class class wbr 5148  (class class class)co 7431  *cxr 11292  cle 11294  [,]cicc 13387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-pre-lttri 11227  ax-pre-lttrn 11228
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-so 5598  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-icc 13391
This theorem is referenced by:  ordtresticc  23247  iccconn  24866  icccvx  24995  oprpiece1res1  24996  oprpiece1res2  24997  pcoass  25071  dvlip  26047  c1liplem1  26050  dvgt0lem1  26056  ftc2ditglem  26101  ttgcontlem1  28914  unitssxrge0  33861  xrge0iifhmeo  33897
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