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Theorem seposep 48722
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 48720. The relationship between separatedness and closure is also seen in isnrm 23359, isnrm2 23382, isnrm3 23383. (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 1208 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑆𝑛)
4 simp1 1135 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝐽 ∈ Top)
5 simp2l 1198 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑛𝐽)
6 eqid 2735 . . . . . . . 8 𝐽 = 𝐽
76eltopss 22929 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑛𝐽) → 𝑛 𝐽)
84, 5, 7syl2anc 584 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑛 𝐽)
93, 8sstrd 4006 . . . . 5 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑆 𝐽)
10 simp32 1209 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑇𝑚)
11 simp2r 1199 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑚𝐽)
126eltopss 22929 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑚𝐽) → 𝑚 𝐽)
134, 11, 12syl2anc 584 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑚 𝐽)
1410, 13sstrd 4006 . . . . 5 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑇 𝐽)
156opncld 23057 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑛𝐽) → ( 𝐽𝑛) ∈ (Clsd‘𝐽))
164, 5, 15syl2anc 584 . . . . . . . . 9 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → ( 𝐽𝑛) ∈ (Clsd‘𝐽))
17 incom 4217 . . . . . . . . . . . 12 (𝑛𝑚) = (𝑚𝑛)
18 simp33 1210 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → (𝑛𝑚) = ∅)
1917, 18eqtr3id 2789 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → (𝑚𝑛) = ∅)
20 reldisj 4459 . . . . . . . . . . . 12 (𝑚 𝐽 → ((𝑚𝑛) = ∅ ↔ 𝑚 ⊆ ( 𝐽𝑛)))
2120biimpd 229 . . . . . . . . . . 11 (𝑚 𝐽 → ((𝑚𝑛) = ∅ → 𝑚 ⊆ ( 𝐽𝑛)))
2213, 19, 21sylc 65 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑚 ⊆ ( 𝐽𝑛))
2310, 22sstrd 4006 . . . . . . . . 9 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑇 ⊆ ( 𝐽𝑛))
246clsss2 23096 . . . . . . . . 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 4156 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → ( 𝐽𝑛) ⊆ ( 𝐽𝑆))
2725, 26sstrd 4006 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → ((cls‘𝐽)‘𝑇) ⊆ ( 𝐽𝑆))
28 disjdif 4478 . . . . . . . 8 (𝑆 ∩ ( 𝐽𝑆)) = ∅
2928a1i 11 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → (𝑆 ∩ ( 𝐽𝑆)) = ∅)
3027, 29ssdisjdr 48657 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → (𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅)
316opncld 23057 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑚𝐽) → ( 𝐽𝑚) ∈ (Clsd‘𝐽))
324, 11, 31syl2anc 584 . . . . . . . . 9 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → ( 𝐽𝑚) ∈ (Clsd‘𝐽))
33 reldisj 4459 . . . . . . . . . . . 12 (𝑛 𝐽 → ((𝑛𝑚) = ∅ ↔ 𝑛 ⊆ ( 𝐽𝑚)))
3433biimpd 229 . . . . . . . . . . 11 (𝑛 𝐽 → ((𝑛𝑚) = ∅ → 𝑛 ⊆ ( 𝐽𝑚)))
358, 18, 34sylc 65 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑛 ⊆ ( 𝐽𝑚))
363, 35sstrd 4006 . . . . . . . . 9 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑆 ⊆ ( 𝐽𝑚))
376clsss2 23096 . . . . . . . . 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 4156 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → ( 𝐽𝑚) ⊆ ( 𝐽𝑇))
4038, 39sstrd 4006 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → ((cls‘𝐽)‘𝑆) ⊆ ( 𝐽𝑇))
41 disjdifr 4479 . . . . . . . 8 (( 𝐽𝑇) ∩ 𝑇) = ∅
4241a1i 11 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → (( 𝐽𝑇) ∩ 𝑇) = ∅)
4340, 42ssdisjd 48656 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)
4430, 43jca 511 . . . . 5 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅))
459, 14, 44jca31 514 . . . 4 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → ((𝑆 𝐽𝑇 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)))
46453exp 1118 . . 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 395  w3a 1086   = wceq 1537  wcel 2106  wrex 3068  cdif 3960  cin 3962  wss 3963  c0 4339   cuni 4912  cfv 6563  Topctop 22915  Clsdccld 23040  clsccl 23042
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-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  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-ral 3060  df-rex 3069  df-reu 3379  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-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  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-top 22916  df-cld 23043  df-cls 23045
This theorem is referenced by: (None)
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