Theorem isnrm3 23342

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 23319	. . 3 ⊢ (𝐽 ∈ Nrm → 𝐽 ∈ Top)
2		nrmsep 23340	. . . . . 6 ⊢ ((𝐽 ∈ Nrm ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑑 ∈ (Clsd‘𝐽) ∧ (𝑐 ∩ 𝑑) = ∅)) → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))
3	2	3exp2 1361	. . . . 5 ⊢ (𝐽 ∈ Nrm → (𝑐 ∈ (Clsd‘𝐽) → (𝑑 ∈ (Clsd‘𝐽) → ((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))))
4	3	impd 411	. . . 4 ⊢ (𝐽 ∈ Nrm → ((𝑐 ∈ (Clsd‘𝐽) ∧ 𝑑 ∈ (Clsd‘𝐽)) → ((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))))
5	4	ralrimivv 3180	. . 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 516	. 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 483	. . 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 1201	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝑐 ⊆ 𝑥)
9		simpr2 1202	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝑑 ⊆ 𝑦)
10		sslin 4171	. . . . . . . . . . . 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 2739	. . . . . . . . . . . . . 14 ⊢ ∪ 𝐽 = ∪ 𝐽
13	12	opncld 23016	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑦) ∈ (Clsd‘𝐽))
14	13	ad4ant13 757	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → (∪ 𝐽 ∖ 𝑦) ∈ (Clsd‘𝐽))
15		simpr3 1203	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → (𝑥 ∩ 𝑦) = ∅)
16		simpllr 781	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝑥 ∈ 𝐽)
17		elssuni 4869	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐽 → 𝑥 ⊆ ∪ 𝐽)
18		reldisj 4381	. . . . . . . . . . . . . 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 233	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝑥 ⊆ (∪ 𝐽 ∖ 𝑦))
21	12	clsss2 23055	. . . . . . . . . . . . 13 ⊢ (((∪ 𝐽 ∖ 𝑦) ∈ (Clsd‘𝐽) ∧ 𝑥 ⊆ (∪ 𝐽 ∖ 𝑦)) → ((cls‘𝐽)‘𝑥) ⊆ (∪ 𝐽 ∖ 𝑦))
22		ssdifin0 4413	. . . . . . . . . . . . 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 590	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → (((cls‘𝐽)‘𝑥) ∩ 𝑦) = ∅)
25		sseq0 4331	. . . . . . . . . . 11 ⊢ (((((cls‘𝐽)‘𝑥) ∩ 𝑑) ⊆ (((cls‘𝐽)‘𝑥) ∩ 𝑦) ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑦) = ∅) → (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅)
26	11, 24, 25	syl2anc 590	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅)
27	8, 26	jca 516	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → (𝑐 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅))
28	27	rexlimdva2 3142	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → (∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑐 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅)))
29	28	reximdva 3152	. . . . . . 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 3153	. . . . 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 3153	. . . 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 407	. . 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 23341	. . 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 589	. 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 210	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 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3053  ∃wrex 3063   ∖ cdif 3880   ∩ cin 3882   ⊆ wss 3883  ∅c0 4261  ∪ cuni 4838  ‘cfv 6485  Topctop 22876  Clsdccld 22999  clsccl 23001  Nrmcnrm 23293

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-iin 4924  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-top 22877  df-cld 23002  df-cls 23004  df-nrm 23300

This theorem is referenced by:  metnrm  24846  isnrm4  49421  dfnrm2  49422  iscnrm3  49442