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Theorem ixxssixx 12745
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)})
21elmpocl 7377 . . 3 (𝑤 ∈ (𝐴𝑂𝐵) → (𝐴 ∈ ℝ*𝐵 ∈ ℝ*))
3 simp1 1133 . . . . . 6 ((𝑤 ∈ ℝ*𝐴𝑅𝑤𝑤𝑆𝐵) → 𝑤 ∈ ℝ*)
43a1i 11 . . . . 5 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝑤 ∈ ℝ*𝐴𝑅𝑤𝑤𝑆𝐵) → 𝑤 ∈ ℝ*))
5 simpl 486 . . . . . 6 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → 𝐴 ∈ ℝ*)
6 3simpa 1145 . . . . . 6 ((𝑤 ∈ ℝ*𝐴𝑅𝑤𝑤𝑆𝐵) → (𝑤 ∈ ℝ*𝐴𝑅𝑤))
7 ixx.3 . . . . . . 7 ((𝐴 ∈ ℝ*𝑤 ∈ ℝ*) → (𝐴𝑅𝑤𝐴𝑇𝑤))
87expimpd 457 . . . . . 6 (𝐴 ∈ ℝ* → ((𝑤 ∈ ℝ*𝐴𝑅𝑤) → 𝐴𝑇𝑤))
95, 6, 8syl2im 40 . . . . 5 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝑤 ∈ ℝ*𝐴𝑅𝑤𝑤𝑆𝐵) → 𝐴𝑇𝑤))
10 simpr 488 . . . . . 6 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → 𝐵 ∈ ℝ*)
11 3simpb 1146 . . . . . 6 ((𝑤 ∈ ℝ*𝐴𝑅𝑤𝑤𝑆𝐵) → (𝑤 ∈ ℝ*𝑤𝑆𝐵))
12 ixx.4 . . . . . . . 8 ((𝑤 ∈ ℝ*𝐵 ∈ ℝ*) → (𝑤𝑆𝐵𝑤𝑈𝐵))
1312ancoms 462 . . . . . . 7 ((𝐵 ∈ ℝ*𝑤 ∈ ℝ*) → (𝑤𝑆𝐵𝑤𝑈𝐵))
1413expimpd 457 . . . . . 6 (𝐵 ∈ ℝ* → ((𝑤 ∈ ℝ*𝑤𝑆𝐵) → 𝑤𝑈𝐵))
1510, 11, 14syl2im 40 . . . . 5 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝑤 ∈ ℝ*𝐴𝑅𝑤𝑤𝑆𝐵) → 𝑤𝑈𝐵))
164, 9, 153jcad 1126 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝑤 ∈ ℝ*𝐴𝑅𝑤𝑤𝑆𝐵) → (𝑤 ∈ ℝ*𝐴𝑇𝑤𝑤𝑈𝐵)))
171elixx1 12740 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝑤 ∈ (𝐴𝑂𝐵) ↔ (𝑤 ∈ ℝ*𝐴𝑅𝑤𝑤𝑆𝐵)))
18 ixx.2 . . . . 5 𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑇𝑧𝑧𝑈𝑦)})
1918elixx1 12740 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝑤 ∈ (𝐴𝑃𝐵) ↔ (𝑤 ∈ ℝ*𝐴𝑇𝑤𝑤𝑈𝐵)))
2016, 17, 193imtr4d 297 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝑤 ∈ (𝐴𝑂𝐵) → 𝑤 ∈ (𝐴𝑃𝐵)))
212, 20mpcom 38 . 2 (𝑤 ∈ (𝐴𝑂𝐵) → 𝑤 ∈ (𝐴𝑃𝐵))
2221ssriv 3956 1 (𝐴𝑂𝐵) ⊆ (𝐴𝑃𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  {crab 3137  wss 3919   class class class wbr 5052  (class class class)co 7145  cmpo 7147  *cxr 10666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7451  ax-cnex 10585  ax-resscn 10586
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-iota 6302  df-fun 6345  df-fv 6351  df-ov 7148  df-oprab 7149  df-mpo 7150  df-xr 10671
This theorem is referenced by:  ioossicc  12816  icossicc  12819  iocssicc  12820  ioossico  12821  dvloglem  25235  ioossioc  41992
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