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Theorem iscnrm3 48856
Description: A completely normal topology is a topology in which two separated sets can be separated by open neighborhoods. (Contributed by Zhi Wang, 5-Sep-2024.)
Assertion
Ref Expression
iscnrm3 (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑠 ∈ 𝒫 𝐽𝑡 ∈ 𝒫 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛𝐽𝑚𝐽 (𝑠𝑛𝑡𝑚 ∧ (𝑛𝑚) = ∅))))
Distinct variable group:   𝑚,𝐽,𝑛,𝑠,𝑡

Proof of Theorem iscnrm3
Dummy variables 𝑐 𝑑 𝑘 𝑙 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2736 . . 3 𝐽 = 𝐽
21iscnrm 23332 . 2 (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑧 ∈ 𝒫 𝐽(𝐽t 𝑧) ∈ Nrm))
3 iscnrm3lem1 48838 . . . . 5 (𝐽 ∈ Top → (∀𝑧 ∈ 𝒫 𝐽𝑐 ∈ (Clsd‘(𝐽t 𝑧))∀𝑑 ∈ (Clsd‘(𝐽t 𝑧))((𝑐𝑑) = ∅ → ∃𝑙 ∈ (𝐽t 𝑧)∃𝑘 ∈ (𝐽t 𝑧)(𝑐𝑙𝑑𝑘 ∧ (𝑙𝑘) = ∅)) ↔ ∀𝑧 ∈ 𝒫 𝐽((𝐽t 𝑧) ∈ Top ∧ ∀𝑐 ∈ (Clsd‘(𝐽t 𝑧))∀𝑑 ∈ (Clsd‘(𝐽t 𝑧))((𝑐𝑑) = ∅ → ∃𝑙 ∈ (𝐽t 𝑧)∃𝑘 ∈ (𝐽t 𝑧)(𝑐𝑙𝑑𝑘 ∧ (𝑙𝑘) = ∅)))))
4 isnrm3 23368 . . . . . 6 ((𝐽t 𝑧) ∈ Nrm ↔ ((𝐽t 𝑧) ∈ Top ∧ ∀𝑐 ∈ (Clsd‘(𝐽t 𝑧))∀𝑑 ∈ (Clsd‘(𝐽t 𝑧))((𝑐𝑑) = ∅ → ∃𝑙 ∈ (𝐽t 𝑧)∃𝑘 ∈ (𝐽t 𝑧)(𝑐𝑙𝑑𝑘 ∧ (𝑙𝑘) = ∅))))
54ralbii 3092 . . . . 5 (∀𝑧 ∈ 𝒫 𝐽(𝐽t 𝑧) ∈ Nrm ↔ ∀𝑧 ∈ 𝒫 𝐽((𝐽t 𝑧) ∈ Top ∧ ∀𝑐 ∈ (Clsd‘(𝐽t 𝑧))∀𝑑 ∈ (Clsd‘(𝐽t 𝑧))((𝑐𝑑) = ∅ → ∃𝑙 ∈ (𝐽t 𝑧)∃𝑘 ∈ (𝐽t 𝑧)(𝑐𝑙𝑑𝑘 ∧ (𝑙𝑘) = ∅))))
63, 5bitr4di 289 . . . 4 (𝐽 ∈ Top → (∀𝑧 ∈ 𝒫 𝐽𝑐 ∈ (Clsd‘(𝐽t 𝑧))∀𝑑 ∈ (Clsd‘(𝐽t 𝑧))((𝑐𝑑) = ∅ → ∃𝑙 ∈ (𝐽t 𝑧)∃𝑘 ∈ (𝐽t 𝑧)(𝑐𝑙𝑑𝑘 ∧ (𝑙𝑘) = ∅)) ↔ ∀𝑧 ∈ 𝒫 𝐽(𝐽t 𝑧) ∈ Nrm))
7 iscnrm3r 48852 . . . . 5 (𝐽 ∈ Top → (∀𝑧 ∈ 𝒫 𝐽𝑐 ∈ (Clsd‘(𝐽t 𝑧))∀𝑑 ∈ (Clsd‘(𝐽t 𝑧))((𝑐𝑑) = ∅ → ∃𝑙 ∈ (𝐽t 𝑧)∃𝑘 ∈ (𝐽t 𝑧)(𝑐𝑙𝑑𝑘 ∧ (𝑙𝑘) = ∅)) → ((𝑠 ∈ 𝒫 𝐽𝑡 ∈ 𝒫 𝐽) → (((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛𝐽𝑚𝐽 (𝑠𝑛𝑡𝑚 ∧ (𝑛𝑚) = ∅)))))
8 iscnrm3l 48855 . . . . 5 (𝐽 ∈ Top → (∀𝑠 ∈ 𝒫 𝐽𝑡 ∈ 𝒫 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛𝐽𝑚𝐽 (𝑠𝑛𝑡𝑚 ∧ (𝑛𝑚) = ∅)) → ((𝑧 ∈ 𝒫 𝐽𝑐 ∈ (Clsd‘(𝐽t 𝑧)) ∧ 𝑑 ∈ (Clsd‘(𝐽t 𝑧))) → ((𝑐𝑑) = ∅ → ∃𝑙 ∈ (𝐽t 𝑧)∃𝑘 ∈ (𝐽t 𝑧)(𝑐𝑙𝑑𝑘 ∧ (𝑙𝑘) = ∅)))))
97, 8iscnrm3lem2 48839 . . . 4 (𝐽 ∈ Top → (∀𝑧 ∈ 𝒫 𝐽𝑐 ∈ (Clsd‘(𝐽t 𝑧))∀𝑑 ∈ (Clsd‘(𝐽t 𝑧))((𝑐𝑑) = ∅ → ∃𝑙 ∈ (𝐽t 𝑧)∃𝑘 ∈ (𝐽t 𝑧)(𝑐𝑙𝑑𝑘 ∧ (𝑙𝑘) = ∅)) ↔ ∀𝑠 ∈ 𝒫 𝐽𝑡 ∈ 𝒫 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛𝐽𝑚𝐽 (𝑠𝑛𝑡𝑚 ∧ (𝑛𝑚) = ∅))))
106, 9bitr3d 281 . . 3 (𝐽 ∈ Top → (∀𝑧 ∈ 𝒫 𝐽(𝐽t 𝑧) ∈ Nrm ↔ ∀𝑠 ∈ 𝒫 𝐽𝑡 ∈ 𝒫 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛𝐽𝑚𝐽 (𝑠𝑛𝑡𝑚 ∧ (𝑛𝑚) = ∅))))
1110pm5.32i 574 . 2 ((𝐽 ∈ Top ∧ ∀𝑧 ∈ 𝒫 𝐽(𝐽t 𝑧) ∈ Nrm) ↔ (𝐽 ∈ Top ∧ ∀𝑠 ∈ 𝒫 𝐽𝑡 ∈ 𝒫 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛𝐽𝑚𝐽 (𝑠𝑛𝑡𝑚 ∧ (𝑛𝑚) = ∅))))
122, 11bitri 275 1 (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑠 ∈ 𝒫 𝐽𝑡 ∈ 𝒫 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛𝐽𝑚𝐽 (𝑠𝑛𝑡𝑚 ∧ (𝑛𝑚) = ∅))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  wral 3060  wrex 3069  cin 3949  wss 3950  c0 4332  𝒫 cpw 4599   cuni 4906  cfv 6560  (class class class)co 7432  t crest 17466  Topctop 22900  Clsdccld 23025  clsccl 23027  Nrmcnrm 23319  CNrmccnrm 23320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-en 8987  df-fin 8990  df-fi 9452  df-rest 17468  df-topgen 17489  df-top 22901  df-topon 22918  df-bases 22954  df-cld 23028  df-cls 23030  df-nrm 23326  df-cnrm 23327
This theorem is referenced by:  iscnrm3v  48857  iscnrm4  48858
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