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Theorem ixxssixx 12227
Description: An interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
Hypotheses
Ref Expression
ixx.1 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})
ixx.2 𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑇𝑧𝑧𝑈𝑦)})
ixx.3 ((𝐴 ∈ ℝ*𝑤 ∈ ℝ*) → (𝐴𝑅𝑤𝐴𝑇𝑤))
ixx.4 ((𝑤 ∈ ℝ*𝐵 ∈ ℝ*) → (𝑤𝑆𝐵𝑤𝑈𝐵))
Assertion
Ref Expression
ixxssixx (𝐴𝑂𝐵) ⊆ (𝐴𝑃𝐵)
Distinct variable groups:   𝑥,𝑤,𝑦,𝑧,𝐴   𝑤,𝑂   𝑤,𝐵,𝑥,𝑦,𝑧   𝑤,𝑃   𝑥,𝑅,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝑥,𝑇,𝑦,𝑧   𝑥,𝑈,𝑦,𝑧
Allowed substitution hints:   𝑃(𝑥,𝑦,𝑧)   𝑅(𝑤)   𝑆(𝑤)   𝑇(𝑤)   𝑈(𝑤)   𝑂(𝑥,𝑦,𝑧)

Proof of Theorem ixxssixx
StepHypRef Expression
1 ixx.1 . . . 4 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})
21elmpt2cl 6918 . . 3 (𝑤 ∈ (𝐴𝑂𝐵) → (𝐴 ∈ ℝ*𝐵 ∈ ℝ*))
3 simp1 1081 . . . . . 6 ((𝑤 ∈ ℝ*𝐴𝑅𝑤𝑤𝑆𝐵) → 𝑤 ∈ ℝ*)
43a1i 11 . . . . 5 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝑤 ∈ ℝ*𝐴𝑅𝑤𝑤𝑆𝐵) → 𝑤 ∈ ℝ*))
5 simpl 472 . . . . . 6 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → 𝐴 ∈ ℝ*)
6 3simpa 1078 . . . . . 6 ((𝑤 ∈ ℝ*𝐴𝑅𝑤𝑤𝑆𝐵) → (𝑤 ∈ ℝ*𝐴𝑅𝑤))
7 ixx.3 . . . . . . 7 ((𝐴 ∈ ℝ*𝑤 ∈ ℝ*) → (𝐴𝑅𝑤𝐴𝑇𝑤))
87expimpd 628 . . . . . 6 (𝐴 ∈ ℝ* → ((𝑤 ∈ ℝ*𝐴𝑅𝑤) → 𝐴𝑇𝑤))
95, 6, 8syl2im 40 . . . . 5 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝑤 ∈ ℝ*𝐴𝑅𝑤𝑤𝑆𝐵) → 𝐴𝑇𝑤))
10 simpr 476 . . . . . 6 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → 𝐵 ∈ ℝ*)
11 3simpb 1079 . . . . . 6 ((𝑤 ∈ ℝ*𝐴𝑅𝑤𝑤𝑆𝐵) → (𝑤 ∈ ℝ*𝑤𝑆𝐵))
12 ixx.4 . . . . . . . 8 ((𝑤 ∈ ℝ*𝐵 ∈ ℝ*) → (𝑤𝑆𝐵𝑤𝑈𝐵))
1312ancoms 468 . . . . . . 7 ((𝐵 ∈ ℝ*𝑤 ∈ ℝ*) → (𝑤𝑆𝐵𝑤𝑈𝐵))
1413expimpd 628 . . . . . 6 (𝐵 ∈ ℝ* → ((𝑤 ∈ ℝ*𝑤𝑆𝐵) → 𝑤𝑈𝐵))
1510, 11, 14syl2im 40 . . . . 5 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝑤 ∈ ℝ*𝐴𝑅𝑤𝑤𝑆𝐵) → 𝑤𝑈𝐵))
164, 9, 153jcad 1262 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝑤 ∈ ℝ*𝐴𝑅𝑤𝑤𝑆𝐵) → (𝑤 ∈ ℝ*𝐴𝑇𝑤𝑤𝑈𝐵)))
171elixx1 12222 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝑤 ∈ (𝐴𝑂𝐵) ↔ (𝑤 ∈ ℝ*𝐴𝑅𝑤𝑤𝑆𝐵)))
18 ixx.2 . . . . 5 𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑇𝑧𝑧𝑈𝑦)})
1918elixx1 12222 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝑤 ∈ (𝐴𝑃𝐵) ↔ (𝑤 ∈ ℝ*𝐴𝑇𝑤𝑤𝑈𝐵)))
2016, 17, 193imtr4d 283 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝑤 ∈ (𝐴𝑂𝐵) → 𝑤 ∈ (𝐴𝑃𝐵)))
212, 20mpcom 38 . 2 (𝑤 ∈ (𝐴𝑂𝐵) → 𝑤 ∈ (𝐴𝑃𝐵))
2221ssriv 3640 1 (𝐴𝑂𝐵) ⊆ (𝐴𝑃𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  {crab 2945  wss 3607   class class class wbr 4685  (class class class)co 6690  cmpt2 6692  *cxr 10111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-xr 10116
This theorem is referenced by:  ioossicc  12297  icossicc  12298  iocssicc  12299  ioossico  12300  dvloglem  24439  ioossioc  40031
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