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Theorem seposep 47558
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 47556. The relationship between separatedness and closure is also seen in isnrm 22839, isnrm2 22862, isnrm3 22863. (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 1210 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑆𝑛)
4 simp1 1137 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝐽 ∈ Top)
5 simp2l 1200 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑛𝐽)
6 eqid 2733 . . . . . . . 8 𝐽 = 𝐽
76eltopss 22409 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑛𝐽) → 𝑛 𝐽)
84, 5, 7syl2anc 585 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑛 𝐽)
93, 8sstrd 3993 . . . . 5 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑆 𝐽)
10 simp32 1211 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑇𝑚)
11 simp2r 1201 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑚𝐽)
126eltopss 22409 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑚𝐽) → 𝑚 𝐽)
134, 11, 12syl2anc 585 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑚 𝐽)
1410, 13sstrd 3993 . . . . 5 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑇 𝐽)
156opncld 22537 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑛𝐽) → ( 𝐽𝑛) ∈ (Clsd‘𝐽))
164, 5, 15syl2anc 585 . . . . . . . . 9 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → ( 𝐽𝑛) ∈ (Clsd‘𝐽))
17 incom 4202 . . . . . . . . . . . 12 (𝑛𝑚) = (𝑚𝑛)
18 simp33 1212 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → (𝑛𝑚) = ∅)
1917, 18eqtr3id 2787 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → (𝑚𝑛) = ∅)
20 reldisj 4452 . . . . . . . . . . . 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 3993 . . . . . . . . 9 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑇 ⊆ ( 𝐽𝑛))
246clsss2 22576 . . . . . . . . 9 ((( 𝐽𝑛) ∈ (Clsd‘𝐽) ∧ 𝑇 ⊆ ( 𝐽𝑛)) → ((cls‘𝐽)‘𝑇) ⊆ ( 𝐽𝑛))
2516, 23, 24syl2anc 585 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → ((cls‘𝐽)‘𝑇) ⊆ ( 𝐽𝑛))
263sscond 4142 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → ( 𝐽𝑛) ⊆ ( 𝐽𝑆))
2725, 26sstrd 3993 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → ((cls‘𝐽)‘𝑇) ⊆ ( 𝐽𝑆))
28 disjdif 4472 . . . . . . . 8 (𝑆 ∩ ( 𝐽𝑆)) = ∅
2928a1i 11 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → (𝑆 ∩ ( 𝐽𝑆)) = ∅)
3027, 29ssdisjdr 47493 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → (𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅)
316opncld 22537 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑚𝐽) → ( 𝐽𝑚) ∈ (Clsd‘𝐽))
324, 11, 31syl2anc 585 . . . . . . . . 9 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → ( 𝐽𝑚) ∈ (Clsd‘𝐽))
33 reldisj 4452 . . . . . . . . . . . 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 3993 . . . . . . . . 9 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑆 ⊆ ( 𝐽𝑚))
376clsss2 22576 . . . . . . . . 9 ((( 𝐽𝑚) ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ ( 𝐽𝑚)) → ((cls‘𝐽)‘𝑆) ⊆ ( 𝐽𝑚))
3832, 36, 37syl2anc 585 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → ((cls‘𝐽)‘𝑆) ⊆ ( 𝐽𝑚))
3910sscond 4142 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → ( 𝐽𝑚) ⊆ ( 𝐽𝑇))
4038, 39sstrd 3993 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → ((cls‘𝐽)‘𝑆) ⊆ ( 𝐽𝑇))
41 disjdifr 4473 . . . . . . . 8 (( 𝐽𝑇) ∩ 𝑇) = ∅
4241a1i 11 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → (( 𝐽𝑇) ∩ 𝑇) = ∅)
4340, 42ssdisjd 47492 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)
4430, 43jca 513 . . . . 5 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅))
459, 14, 44jca31 516 . . . 4 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → ((𝑆 𝐽𝑇 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)))
46453exp 1120 . . 3 (𝐽 ∈ Top → ((𝑛𝐽𝑚𝐽) → ((𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅) → ((𝑆 𝐽𝑇 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)))))
4746rexlimdvv 3211 . 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 397  w3a 1088   = wceq 1542  wcel 2107  wrex 3071  cdif 3946  cin 3948  wss 3949  c0 4323   cuni 4909  cfv 6544  Topctop 22395  Clsdccld 22520  clsccl 22522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-top 22396  df-cld 22523  df-cls 22525
This theorem is referenced by: (None)
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