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Theorem iscnrm4 48553
Description: A completely normal topology is a topology in which two separated sets can be separated by neighborhoods. (Contributed by Zhi Wang, 5-Sep-2024.)
Assertion
Ref Expression
iscnrm4 (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑠 ∈ 𝒫 𝐽𝑡 ∈ 𝒫 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ ((nei‘𝐽)‘𝑠)∃𝑚 ∈ ((nei‘𝐽)‘𝑡)(𝑛𝑚) = ∅)))
Distinct variable group:   𝑚,𝐽,𝑛,𝑠,𝑡

Proof of Theorem iscnrm4
StepHypRef Expression
1 iscnrm3 48551 . 2 (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑠 ∈ 𝒫 𝐽𝑡 ∈ 𝒫 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛𝐽𝑚𝐽 (𝑠𝑛𝑡𝑚 ∧ (𝑛𝑚) = ∅))))
2 id 22 . . . . . 6 (𝐽 ∈ Top → 𝐽 ∈ Top)
32sepnsepo 48522 . . . . 5 (𝐽 ∈ Top → (∃𝑛 ∈ ((nei‘𝐽)‘𝑠)∃𝑚 ∈ ((nei‘𝐽)‘𝑡)(𝑛𝑚) = ∅ ↔ ∃𝑛𝐽𝑚𝐽 (𝑠𝑛𝑡𝑚 ∧ (𝑛𝑚) = ∅)))
43imbi2d 340 . . . 4 (𝐽 ∈ Top → ((((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ ((nei‘𝐽)‘𝑠)∃𝑚 ∈ ((nei‘𝐽)‘𝑡)(𝑛𝑚) = ∅) ↔ (((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛𝐽𝑚𝐽 (𝑠𝑛𝑡𝑚 ∧ (𝑛𝑚) = ∅))))
542ralbidv 3222 . . 3 (𝐽 ∈ Top → (∀𝑠 ∈ 𝒫 𝐽𝑡 ∈ 𝒫 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ ((nei‘𝐽)‘𝑠)∃𝑚 ∈ ((nei‘𝐽)‘𝑡)(𝑛𝑚) = ∅) ↔ ∀𝑠 ∈ 𝒫 𝐽𝑡 ∈ 𝒫 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛𝐽𝑚𝐽 (𝑠𝑛𝑡𝑚 ∧ (𝑛𝑚) = ∅))))
65pm5.32i 574 . 2 ((𝐽 ∈ Top ∧ ∀𝑠 ∈ 𝒫 𝐽𝑡 ∈ 𝒫 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ ((nei‘𝐽)‘𝑠)∃𝑚 ∈ ((nei‘𝐽)‘𝑡)(𝑛𝑚) = ∅)) ↔ (𝐽 ∈ Top ∧ ∀𝑠 ∈ 𝒫 𝐽𝑡 ∈ 𝒫 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛𝐽𝑚𝐽 (𝑠𝑛𝑡𝑚 ∧ (𝑛𝑚) = ∅))))
71, 6bitr4i 278 1 (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑠 ∈ 𝒫 𝐽𝑡 ∈ 𝒫 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ ((nei‘𝐽)‘𝑠)∃𝑚 ∈ ((nei‘𝐽)‘𝑡)(𝑛𝑚) = ∅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1537  wcel 2103  wral 3063  wrex 3072  cin 3969  wss 3970  c0 4347  𝒫 cpw 4622   cuni 4931  cfv 6572  Topctop 22913  clsccl 23040  neicnei 23119  CNrmccnrm 23333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-pss 3990  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4973  df-iun 5021  df-iin 5022  df-br 5170  df-opab 5232  df-mpt 5253  df-tr 5287  df-id 5597  df-eprel 5603  df-po 5611  df-so 5612  df-fr 5654  df-we 5656  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-ord 6397  df-on 6398  df-lim 6399  df-suc 6400  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-ov 7448  df-oprab 7449  df-mpo 7450  df-om 7900  df-1st 8026  df-2nd 8027  df-en 9000  df-fin 9003  df-fi 9476  df-rest 17477  df-topgen 17498  df-top 22914  df-topon 22931  df-bases 22967  df-cld 23041  df-cls 23043  df-nei 23120  df-nrm 23339  df-cnrm 23340
This theorem is referenced by: (None)
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