Theorem iscnrm4 48796

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 ∪ 𝐽∀𝑡 ∈ 𝒫 ∪ 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ ((nei‘𝐽)‘𝑠)∃𝑚 ∈ ((nei‘𝐽)‘𝑡)(𝑛 ∩ 𝑚) = ∅)))

Distinct variable group:   𝑚,𝐽,𝑛,𝑠,𝑡

Proof of Theorem iscnrm4

Step	Hyp	Ref	Expression
1		iscnrm3 48794	. 2 ⊢ (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑠 ∈ 𝒫 ∪ 𝐽∀𝑡 ∈ 𝒫 ∪ 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))))
2		id 22	. . . . . 6 ⊢ (𝐽 ∈ Top → 𝐽 ∈ Top)
3	2	sepnsepo 48765	. . . . 5 ⊢ (𝐽 ∈ Top → (∃𝑛 ∈ ((nei‘𝐽)‘𝑠)∃𝑚 ∈ ((nei‘𝐽)‘𝑡)(𝑛 ∩ 𝑚) = ∅ ↔ ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)))
4	3	imbi2d 340	. . . 4 ⊢ (𝐽 ∈ Top → ((((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ ((nei‘𝐽)‘𝑠)∃𝑚 ∈ ((nei‘𝐽)‘𝑡)(𝑛 ∩ 𝑚) = ∅) ↔ (((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))))
5	4	2ralbidv 3221	. . 3 ⊢ (𝐽 ∈ Top → (∀𝑠 ∈ 𝒫 ∪ 𝐽∀𝑡 ∈ 𝒫 ∪ 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ ((nei‘𝐽)‘𝑠)∃𝑚 ∈ ((nei‘𝐽)‘𝑡)(𝑛 ∩ 𝑚) = ∅) ↔ ∀𝑠 ∈ 𝒫 ∪ 𝐽∀𝑡 ∈ 𝒫 ∪ 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))))
6	5	pm5.32i 574	. 2 ⊢ ((𝐽 ∈ Top ∧ ∀𝑠 ∈ 𝒫 ∪ 𝐽∀𝑡 ∈ 𝒫 ∪ 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ ((nei‘𝐽)‘𝑠)∃𝑚 ∈ ((nei‘𝐽)‘𝑡)(𝑛 ∩ 𝑚) = ∅)) ↔ (𝐽 ∈ Top ∧ ∀𝑠 ∈ 𝒫 ∪ 𝐽∀𝑡 ∈ 𝒫 ∪ 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))))
7	1, 6	bitr4i 278	1 ⊢ (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑠 ∈ 𝒫 ∪ 𝐽∀𝑡 ∈ 𝒫 ∪ 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ ((nei‘𝐽)‘𝑠)∃𝑚 ∈ ((nei‘𝐽)‘𝑡)(𝑛 ∩ 𝑚) = ∅)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1539   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070   ∩ cin 3965   ⊆ wss 3966  ∅c0 4342  𝒫 cpw 4608  ∪ cuni 4915  ‘cfv 6569  Topctop 22924  clsccl 23051  neicnei 23130  CNrmccnrm 23344

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 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 5288  ax-sep 5305  ax-nul 5315  ax-pow 5374  ax-pr 5441  ax-un 7761

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  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 3483  df-sbc 3795  df-csb 3912  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-pss 3986  df-nul 4343  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-int 4955  df-iun 5001  df-iin 5002  df-br 5152  df-opab 5214  df-mpt 5235  df-tr 5269  df-id 5587  df-eprel 5593  df-po 5601  df-so 5602  df-fr 5645  df-we 5647  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-rn 5704  df-res 5705  df-ima 5706  df-ord 6395  df-on 6396  df-lim 6397  df-suc 6398  df-iota 6522  df-fun 6571  df-fn 6572  df-f 6573  df-f1 6574  df-fo 6575  df-f1o 6576  df-fv 6577  df-ov 7441  df-oprab 7442  df-mpo 7443  df-om 7895  df-1st 8022  df-2nd 8023  df-en 8994  df-fin 8997  df-fi 9458  df-rest 17478  df-topgen 17499  df-top 22925  df-topon 22942  df-bases 22978  df-cld 23052  df-cls 23054  df-nei 23131  df-nrm 23350  df-cnrm 23351

This theorem is referenced by: (None)