Theorem iscnrm3 48749

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

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.

Step	Hyp	Ref	Expression
1		eqid 2735	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	iscnrm 23347	. 2 ⊢ (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑧 ∈ 𝒫 ∪ 𝐽(𝐽 ↾t 𝑧) ∈ Nrm))
3		iscnrm3lem1 48730	. . . . 5 ⊢ (𝐽 ∈ Top → (∀𝑧 ∈ 𝒫 ∪ 𝐽∀𝑐 ∈ (Clsd‘(𝐽 ↾t 𝑧))∀𝑑 ∈ (Clsd‘(𝐽 ↾t 𝑧))((𝑐 ∩ 𝑑) = ∅ → ∃𝑙 ∈ (𝐽 ↾t 𝑧)∃𝑘 ∈ (𝐽 ↾t 𝑧)(𝑐 ⊆ 𝑙 ∧ 𝑑 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)) ↔ ∀𝑧 ∈ 𝒫 ∪ 𝐽((𝐽 ↾t 𝑧) ∈ Top ∧ ∀𝑐 ∈ (Clsd‘(𝐽 ↾t 𝑧))∀𝑑 ∈ (Clsd‘(𝐽 ↾t 𝑧))((𝑐 ∩ 𝑑) = ∅ → ∃𝑙 ∈ (𝐽 ↾t 𝑧)∃𝑘 ∈ (𝐽 ↾t 𝑧)(𝑐 ⊆ 𝑙 ∧ 𝑑 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)))))
4		isnrm3 23383	. . . . . 6 ⊢ ((𝐽 ↾t 𝑧) ∈ Nrm ↔ ((𝐽 ↾t 𝑧) ∈ Top ∧ ∀𝑐 ∈ (Clsd‘(𝐽 ↾t 𝑧))∀𝑑 ∈ (Clsd‘(𝐽 ↾t 𝑧))((𝑐 ∩ 𝑑) = ∅ → ∃𝑙 ∈ (𝐽 ↾t 𝑧)∃𝑘 ∈ (𝐽 ↾t 𝑧)(𝑐 ⊆ 𝑙 ∧ 𝑑 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅))))
5	4	ralbii 3091	. . . . 5 ⊢ (∀𝑧 ∈ 𝒫 ∪ 𝐽(𝐽 ↾t 𝑧) ∈ Nrm ↔ ∀𝑧 ∈ 𝒫 ∪ 𝐽((𝐽 ↾t 𝑧) ∈ Top ∧ ∀𝑐 ∈ (Clsd‘(𝐽 ↾t 𝑧))∀𝑑 ∈ (Clsd‘(𝐽 ↾t 𝑧))((𝑐 ∩ 𝑑) = ∅ → ∃𝑙 ∈ (𝐽 ↾t 𝑧)∃𝑘 ∈ (𝐽 ↾t 𝑧)(𝑐 ⊆ 𝑙 ∧ 𝑑 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅))))
6	3, 5	bitr4di 289	. . . 4 ⊢ (𝐽 ∈ Top → (∀𝑧 ∈ 𝒫 ∪ 𝐽∀𝑐 ∈ (Clsd‘(𝐽 ↾t 𝑧))∀𝑑 ∈ (Clsd‘(𝐽 ↾t 𝑧))((𝑐 ∩ 𝑑) = ∅ → ∃𝑙 ∈ (𝐽 ↾t 𝑧)∃𝑘 ∈ (𝐽 ↾t 𝑧)(𝑐 ⊆ 𝑙 ∧ 𝑑 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)) ↔ ∀𝑧 ∈ 𝒫 ∪ 𝐽(𝐽 ↾t 𝑧) ∈ Nrm))
7		iscnrm3r 48745	. . . . 5 ⊢ (𝐽 ∈ Top → (∀𝑧 ∈ 𝒫 ∪ 𝐽∀𝑐 ∈ (Clsd‘(𝐽 ↾t 𝑧))∀𝑑 ∈ (Clsd‘(𝐽 ↾t 𝑧))((𝑐 ∩ 𝑑) = ∅ → ∃𝑙 ∈ (𝐽 ↾t 𝑧)∃𝑘 ∈ (𝐽 ↾t 𝑧)(𝑐 ⊆ 𝑙 ∧ 𝑑 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)) → ((𝑠 ∈ 𝒫 ∪ 𝐽 ∧ 𝑡 ∈ 𝒫 ∪ 𝐽) → (((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)))))
8		iscnrm3l 48748	. . . . 5 ⊢ (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) = ∅)) → ((𝑧 ∈ 𝒫 ∪ 𝐽 ∧ 𝑐 ∈ (Clsd‘(𝐽 ↾t 𝑧)) ∧ 𝑑 ∈ (Clsd‘(𝐽 ↾t 𝑧))) → ((𝑐 ∩ 𝑑) = ∅ → ∃𝑙 ∈ (𝐽 ↾t 𝑧)∃𝑘 ∈ (𝐽 ↾t 𝑧)(𝑐 ⊆ 𝑙 ∧ 𝑑 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)))))
9	7, 8	iscnrm3lem2 48731	. . . 4 ⊢ (𝐽 ∈ Top → (∀𝑧 ∈ 𝒫 ∪ 𝐽∀𝑐 ∈ (Clsd‘(𝐽 ↾t 𝑧))∀𝑑 ∈ (Clsd‘(𝐽 ↾t 𝑧))((𝑐 ∩ 𝑑) = ∅ → ∃𝑙 ∈ (𝐽 ↾t 𝑧)∃𝑘 ∈ (𝐽 ↾t 𝑧)(𝑐 ⊆ 𝑙 ∧ 𝑑 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)) ↔ ∀𝑠 ∈ 𝒫 ∪ 𝐽∀𝑡 ∈ 𝒫 ∪ 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))))
10	6, 9	bitr3d 281	. . 3 ⊢ (𝐽 ∈ Top → (∀𝑧 ∈ 𝒫 ∪ 𝐽(𝐽 ↾t 𝑧) ∈ Nrm ↔ ∀𝑠 ∈ 𝒫 ∪ 𝐽∀𝑡 ∈ 𝒫 ∪ 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))))
11	10	pm5.32i 574	. 2 ⊢ ((𝐽 ∈ Top ∧ ∀𝑧 ∈ 𝒫 ∪ 𝐽(𝐽 ↾t 𝑧) ∈ Nrm) ↔ (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) = ∅))))
12	2, 11	bitri 275	1 ⊢ (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) = ∅))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068   ∩ cin 3962   ⊆ wss 3963  ∅c0 4339  𝒫 cpw 4605  ∪ cuni 4912  ‘cfv 6563  (class class class)co 7431   ↾_t crest 17467  Topctop 22915  Clsdccld 23040  clsccl 23042  Nrmcnrm 23334  CNrmccnrm 23335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-en 8985  df-fin 8988  df-fi 9449  df-rest 17469  df-topgen 17490  df-top 22916  df-topon 22933  df-bases 22969  df-cld 23043  df-cls 23045  df-nrm 23341  df-cnrm 23342

This theorem is referenced by:  iscnrm3v  48750  iscnrm4  48751