Users' Mathboxes Mathbox for Zhi Wang < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  seposep Structured version   Visualization version   GIF version

Theorem seposep 47646
Description: If two sets are separated by (open) neighborhoods, then they are separated subsets of the underlying set. Note that separatedness by open neighborhoods is equivalent to separatedness by neighborhoods. See sepnsepo 47644. The relationship between separatedness and closure is also seen in isnrm 23059, isnrm2 23082, isnrm3 23083. (Contributed by Zhi Wang, 7-Sep-2024.)
Hypotheses
Ref Expression
sepdisj.1 (𝜑𝐽 ∈ Top)
seposep.2 (𝜑 → ∃𝑛𝐽𝑚𝐽 (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅))
Assertion
Ref Expression
seposep (𝜑 → ((𝑆 𝐽𝑇 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)))
Distinct variable groups:   𝑚,𝐽,𝑛   𝑆,𝑚,𝑛   𝑇,𝑚,𝑛
Allowed substitution hints:   𝜑(𝑚,𝑛)

Proof of Theorem seposep
StepHypRef Expression
1 sepdisj.1 . 2 (𝜑𝐽 ∈ Top)
2 seposep.2 . 2 (𝜑 → ∃𝑛𝐽𝑚𝐽 (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅))
3 simp31 1209 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑆𝑛)
4 simp1 1136 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝐽 ∈ Top)
5 simp2l 1199 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑛𝐽)
6 eqid 2732 . . . . . . . 8 𝐽 = 𝐽
76eltopss 22629 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑛𝐽) → 𝑛 𝐽)
84, 5, 7syl2anc 584 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑛 𝐽)
93, 8sstrd 3992 . . . . 5 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑆 𝐽)
10 simp32 1210 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑇𝑚)
11 simp2r 1200 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑚𝐽)
126eltopss 22629 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑚𝐽) → 𝑚 𝐽)
134, 11, 12syl2anc 584 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑚 𝐽)
1410, 13sstrd 3992 . . . . 5 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑇 𝐽)
156opncld 22757 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑛𝐽) → ( 𝐽𝑛) ∈ (Clsd‘𝐽))
164, 5, 15syl2anc 584 . . . . . . . . 9 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → ( 𝐽𝑛) ∈ (Clsd‘𝐽))
17 incom 4201 . . . . . . . . . . . 12 (𝑛𝑚) = (𝑚𝑛)
18 simp33 1211 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → (𝑛𝑚) = ∅)
1917, 18eqtr3id 2786 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → (𝑚𝑛) = ∅)
20 reldisj 4451 . . . . . . . . . . . 12 (𝑚 𝐽 → ((𝑚𝑛) = ∅ ↔ 𝑚 ⊆ ( 𝐽𝑛)))
2120biimpd 228 . . . . . . . . . . 11 (𝑚 𝐽 → ((𝑚𝑛) = ∅ → 𝑚 ⊆ ( 𝐽𝑛)))
2213, 19, 21sylc 65 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑚 ⊆ ( 𝐽𝑛))
2310, 22sstrd 3992 . . . . . . . . 9 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑇 ⊆ ( 𝐽𝑛))
246clsss2 22796 . . . . . . . . 9 ((( 𝐽𝑛) ∈ (Clsd‘𝐽) ∧ 𝑇 ⊆ ( 𝐽𝑛)) → ((cls‘𝐽)‘𝑇) ⊆ ( 𝐽𝑛))
2516, 23, 24syl2anc 584 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → ((cls‘𝐽)‘𝑇) ⊆ ( 𝐽𝑛))
263sscond 4141 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → ( 𝐽𝑛) ⊆ ( 𝐽𝑆))
2725, 26sstrd 3992 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → ((cls‘𝐽)‘𝑇) ⊆ ( 𝐽𝑆))
28 disjdif 4471 . . . . . . . 8 (𝑆 ∩ ( 𝐽𝑆)) = ∅
2928a1i 11 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → (𝑆 ∩ ( 𝐽𝑆)) = ∅)
3027, 29ssdisjdr 47581 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → (𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅)
316opncld 22757 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑚𝐽) → ( 𝐽𝑚) ∈ (Clsd‘𝐽))
324, 11, 31syl2anc 584 . . . . . . . . 9 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → ( 𝐽𝑚) ∈ (Clsd‘𝐽))
33 reldisj 4451 . . . . . . . . . . . 12 (𝑛 𝐽 → ((𝑛𝑚) = ∅ ↔ 𝑛 ⊆ ( 𝐽𝑚)))
3433biimpd 228 . . . . . . . . . . 11 (𝑛 𝐽 → ((𝑛𝑚) = ∅ → 𝑛 ⊆ ( 𝐽𝑚)))
358, 18, 34sylc 65 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑛 ⊆ ( 𝐽𝑚))
363, 35sstrd 3992 . . . . . . . . 9 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑆 ⊆ ( 𝐽𝑚))
376clsss2 22796 . . . . . . . . 9 ((( 𝐽𝑚) ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ ( 𝐽𝑚)) → ((cls‘𝐽)‘𝑆) ⊆ ( 𝐽𝑚))
3832, 36, 37syl2anc 584 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → ((cls‘𝐽)‘𝑆) ⊆ ( 𝐽𝑚))
3910sscond 4141 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → ( 𝐽𝑚) ⊆ ( 𝐽𝑇))
4038, 39sstrd 3992 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → ((cls‘𝐽)‘𝑆) ⊆ ( 𝐽𝑇))
41 disjdifr 4472 . . . . . . . 8 (( 𝐽𝑇) ∩ 𝑇) = ∅
4241a1i 11 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → (( 𝐽𝑇) ∩ 𝑇) = ∅)
4340, 42ssdisjd 47580 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)
4430, 43jca 512 . . . . 5 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅))
459, 14, 44jca31 515 . . . 4 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → ((𝑆 𝐽𝑇 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)))
46453exp 1119 . . 3 (𝐽 ∈ Top → ((𝑛𝐽𝑚𝐽) → ((𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅) → ((𝑆 𝐽𝑇 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)))))
4746rexlimdvv 3210 . 2 (𝐽 ∈ Top → (∃𝑛𝐽𝑚𝐽 (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅) → ((𝑆 𝐽𝑇 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅))))
481, 2, 47sylc 65 1 (𝜑 → ((𝑆 𝐽𝑇 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wrex 3070  cdif 3945  cin 3947  wss 3948  c0 4322   cuni 4908  cfv 6543  Topctop 22615  Clsdccld 22740  clsccl 22742
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-top 22616  df-cld 22743  df-cls 22745
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator