Theorem isnrm3 23302

Description: A topological space is normal iff any two disjoint closed sets are separated by open sets. (Contributed by Mario Carneiro, 24-Aug-2015.)

Assertion

Ref	Expression
isnrm3	⊢ (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))))

Distinct variable groups:   𝑥,𝑦   𝑐,𝑑,𝑥,𝑦,𝐽

Proof of Theorem isnrm3

Step	Hyp	Ref	Expression
1		nrmtop 23279	. . 3 ⊢ (𝐽 ∈ Nrm → 𝐽 ∈ Top)
2		nrmsep 23300	. . . . . 6 ⊢ ((𝐽 ∈ Nrm ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑑 ∈ (Clsd‘𝐽) ∧ (𝑐 ∩ 𝑑) = ∅)) → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))
3	2	3exp2 1355	. . . . 5 ⊢ (𝐽 ∈ Nrm → (𝑐 ∈ (Clsd‘𝐽) → (𝑑 ∈ (Clsd‘𝐽) → ((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))))
4	3	impd 410	. . . 4 ⊢ (𝐽 ∈ Nrm → ((𝑐 ∈ (Clsd‘𝐽) ∧ 𝑑 ∈ (Clsd‘𝐽)) → ((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))))
5	4	ralrimivv 3186	. . 3 ⊢ (𝐽 ∈ Nrm → ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))
6	1, 5	jca 511	. 2 ⊢ (𝐽 ∈ Nrm → (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))))
7		simpl 482	. . 3 ⊢ ((𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))) → 𝐽 ∈ Top)
8		simpr1 1195	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝑐 ⊆ 𝑥)
9		simpr2 1196	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝑑 ⊆ 𝑦)
10		sslin 4223	. . . . . . . . . . . 12 ⊢ (𝑑 ⊆ 𝑦 → (((cls‘𝐽)‘𝑥) ∩ 𝑑) ⊆ (((cls‘𝐽)‘𝑥) ∩ 𝑦))
11	9, 10	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → (((cls‘𝐽)‘𝑥) ∩ 𝑑) ⊆ (((cls‘𝐽)‘𝑥) ∩ 𝑦))
12		eqid 2736	. . . . . . . . . . . . . 14 ⊢ ∪ 𝐽 = ∪ 𝐽
13	12	opncld 22976	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑦) ∈ (Clsd‘𝐽))
14	13	ad4ant13 751	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → (∪ 𝐽 ∖ 𝑦) ∈ (Clsd‘𝐽))
15		simpr3 1197	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → (𝑥 ∩ 𝑦) = ∅)
16		simpllr 775	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝑥 ∈ 𝐽)
17		elssuni 4918	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐽 → 𝑥 ⊆ ∪ 𝐽)
18		reldisj 4433	. . . . . . . . . . . . . 14 ⊢ (𝑥 ⊆ ∪ 𝐽 → ((𝑥 ∩ 𝑦) = ∅ ↔ 𝑥 ⊆ (∪ 𝐽 ∖ 𝑦)))
19	16, 17, 18	3syl 18	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → ((𝑥 ∩ 𝑦) = ∅ ↔ 𝑥 ⊆ (∪ 𝐽 ∖ 𝑦)))
20	15, 19	mpbid 232	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝑥 ⊆ (∪ 𝐽 ∖ 𝑦))
21	12	clsss2 23015	. . . . . . . . . . . . 13 ⊢ (((∪ 𝐽 ∖ 𝑦) ∈ (Clsd‘𝐽) ∧ 𝑥 ⊆ (∪ 𝐽 ∖ 𝑦)) → ((cls‘𝐽)‘𝑥) ⊆ (∪ 𝐽 ∖ 𝑦))
22		ssdifin0 4466	. . . . . . . . . . . . 13 ⊢ (((cls‘𝐽)‘𝑥) ⊆ (∪ 𝐽 ∖ 𝑦) → (((cls‘𝐽)‘𝑥) ∩ 𝑦) = ∅)
23	21, 22	syl 17	. . . . . . . . . . . 12 ⊢ (((∪ 𝐽 ∖ 𝑦) ∈ (Clsd‘𝐽) ∧ 𝑥 ⊆ (∪ 𝐽 ∖ 𝑦)) → (((cls‘𝐽)‘𝑥) ∩ 𝑦) = ∅)
24	14, 20, 23	syl2anc 584	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → (((cls‘𝐽)‘𝑥) ∩ 𝑦) = ∅)
25		sseq0 4383	. . . . . . . . . . 11 ⊢ (((((cls‘𝐽)‘𝑥) ∩ 𝑑) ⊆ (((cls‘𝐽)‘𝑥) ∩ 𝑦) ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑦) = ∅) → (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅)
26	11, 24, 25	syl2anc 584	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅)
27	8, 26	jca 511	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → (𝑐 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅))
28	27	rexlimdva2 3144	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → (∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑐 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅)))
29	28	reximdva 3154	. . . . . . 7 ⊢ (𝐽 ∈ Top → (∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅) → ∃𝑥 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅)))
30	29	imim2d 57	. . . . . 6 ⊢ (𝐽 ∈ Top → (((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → ((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅))))
31	30	ralimdv 3155	. . . . 5 ⊢ (𝐽 ∈ Top → (∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → ∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅))))
32	31	ralimdv 3155	. . . 4 ⊢ (𝐽 ∈ Top → (∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅))))
33	32	imp 406	. . 3 ⊢ ((𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))) → ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅)))
34		isnrm2 23301	. . 3 ⊢ (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅))))
35	7, 33, 34	sylanbrc 583	. 2 ⊢ ((𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))) → 𝐽 ∈ Nrm)
36	6, 35	impbii 209	1 ⊢ (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3052  ∃wrex 3061   ∖ cdif 3928   ∩ cin 3930   ⊆ wss 3931  ∅c0 4313  ∪ cuni 4888  ‘cfv 6536  Topctop 22836  Clsdccld 22959  clsccl 22961  Nrmcnrm 23253

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-iin 4975  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-top 22837  df-cld 22962  df-cls 22964  df-nrm 23260

This theorem is referenced by:  metnrm  24807  isnrm4  48872  dfnrm2  48873  iscnrm3  48893