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Theorem seposep 48823
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 48821. The relationship between separatedness and closure is also seen in isnrm 23343, isnrm2 23366, isnrm3 23367. (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 2737 . . . . . . . 8 𝐽 = 𝐽
76eltopss 22913 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑛𝐽) → 𝑛 𝐽)
84, 5, 7syl2anc 584 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑛 𝐽)
93, 8sstrd 3994 . . . . 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 22913 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑚𝐽) → 𝑚 𝐽)
134, 11, 12syl2anc 584 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑚 𝐽)
1410, 13sstrd 3994 . . . . 5 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑇 𝐽)
156opncld 23041 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑛𝐽) → ( 𝐽𝑛) ∈ (Clsd‘𝐽))
164, 5, 15syl2anc 584 . . . . . . . . 9 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → ( 𝐽𝑛) ∈ (Clsd‘𝐽))
17 incom 4209 . . . . . . . . . . . 12 (𝑛𝑚) = (𝑚𝑛)
18 simp33 1212 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → (𝑛𝑚) = ∅)
1917, 18eqtr3id 2791 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → (𝑚𝑛) = ∅)
20 reldisj 4453 . . . . . . . . . . . 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 3994 . . . . . . . . 9 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑇 ⊆ ( 𝐽𝑛))
246clsss2 23080 . . . . . . . . 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 4146 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → ( 𝐽𝑛) ⊆ ( 𝐽𝑆))
2725, 26sstrd 3994 . . . . . . 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 48728 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → (𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅)
316opncld 23041 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑚𝐽) → ( 𝐽𝑚) ∈ (Clsd‘𝐽))
324, 11, 31syl2anc 584 . . . . . . . . 9 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → ( 𝐽𝑚) ∈ (Clsd‘𝐽))
33 reldisj 4453 . . . . . . . . . . . 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 3994 . . . . . . . . 9 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑆 ⊆ ( 𝐽𝑚))
376clsss2 23080 . . . . . . . . 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 4146 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝑛𝐽𝑚𝐽) ∧ (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)) → ( 𝐽𝑚) ⊆ ( 𝐽𝑇))
4038, 39sstrd 3994 . . . . . . 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 48727 . . . . . 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 1120 . . 3 (𝐽 ∈ Top → ((𝑛𝐽𝑚𝐽) → ((𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅) → ((𝑆 𝐽𝑇 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)))))
4746rexlimdvv 3212 . 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 1087   = wceq 1540  wcel 2108  wrex 3070  cdif 3948  cin 3950  wss 3951  c0 4333   cuni 4907  cfv 6561  Topctop 22899  Clsdccld 23024  clsccl 23026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-top 22900  df-cld 23027  df-cls 23029
This theorem is referenced by: (None)
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