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Theorem iccss2 9888
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 9839 . . . . . 6 [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧𝑦)})
21elixx3g 9845 . . . . 5 (𝐶 ∈ (𝐴[,]𝐵) ↔ ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐴𝐶𝐶𝐵)))
32simplbi 272 . . . 4 (𝐶 ∈ (𝐴[,]𝐵) → (𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*))
43adantr 274 . . 3 ((𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐷 ∈ (𝐴[,]𝐵)) → (𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*))
54simp1d 1004 . 2 ((𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐷 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ*)
64simp2d 1005 . 2 ((𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐷 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ*)
72simprbi 273 . . . 4 (𝐶 ∈ (𝐴[,]𝐵) → (𝐴𝐶𝐶𝐵))
87adantr 274 . . 3 ((𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐷 ∈ (𝐴[,]𝐵)) → (𝐴𝐶𝐶𝐵))
98simpld 111 . 2 ((𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐷 ∈ (𝐴[,]𝐵)) → 𝐴𝐶)
101elixx3g 9845 . . . . 5 (𝐷 ∈ (𝐴[,]𝐵) ↔ ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐷 ∈ ℝ*) ∧ (𝐴𝐷𝐷𝐵)))
1110simprbi 273 . . . 4 (𝐷 ∈ (𝐴[,]𝐵) → (𝐴𝐷𝐷𝐵))
1211simprd 113 . . 3 (𝐷 ∈ (𝐴[,]𝐵) → 𝐷𝐵)
1312adantl 275 . 2 ((𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐷 ∈ (𝐴[,]𝐵)) → 𝐷𝐵)
14 xrletr 9752 . . 3 ((𝐴 ∈ ℝ*𝐶 ∈ ℝ*𝑤 ∈ ℝ*) → ((𝐴𝐶𝐶𝑤) → 𝐴𝑤))
15 xrletr 9752 . . 3 ((𝑤 ∈ ℝ*𝐷 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝑤𝐷𝐷𝐵) → 𝑤𝐵))
161, 1, 14, 15ixxss12 9850 . 2 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐴𝐶𝐷𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,]𝐵))
175, 6, 9, 13, 16syl22anc 1234 1 ((𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐷 ∈ (𝐴[,]𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,]𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 973  wcel 2141  wss 3121   class class class wbr 3987  (class class class)co 5850  *cxr 7940  cle 7942  [,]cicc 9835
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7852  ax-resscn 7853  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-id 4276  df-po 4279  df-iso 4280  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-iota 5158  df-fun 5198  df-fv 5204  df-ov 5853  df-oprab 5854  df-mpo 5855  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-icc 9839
This theorem is referenced by: (None)
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