Theorem nrmsep 21384

Description: In a normal space, disjoint closed sets are separated by open sets. (Contributed by Jeff Hankins, 1-Feb-2010.)

Assertion

Ref	Expression
nrmsep	⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ 𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))

Distinct variable groups:   𝑥,𝑦,𝐶   𝑥,𝐷,𝑦   𝑥,𝐽,𝑦

Proof of Theorem nrmsep

Step	Hyp	Ref	Expression
1		nrmtop 21363	. . . . . 6 ⊢ (𝐽 ∈ Nrm → 𝐽 ∈ Top)
2	1	ad2antrr 764	. . . . 5 ⊢ (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) ∧ (𝑥 ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝐽 ∈ Top)
3		elssuni 4620	. . . . . 6 ⊢ (𝑥 ∈ 𝐽 → 𝑥 ⊆ ∪ 𝐽)
4	3	ad2antrl 766	. . . . 5 ⊢ (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) ∧ (𝑥 ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝑥 ⊆ ∪ 𝐽)
5		eqid 2761	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
6	5	clscld 21074	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽))
7	2, 4, 6	syl2anc 696	. . . 4 ⊢ (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) ∧ (𝑥 ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → ((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽))
8	5	cldopn 21058	. . . 4 ⊢ (((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥)) ∈ 𝐽)
9	7, 8	syl 17	. . 3 ⊢ (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) ∧ (𝑥 ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥)) ∈ 𝐽)
10		simprrl 823	. . 3 ⊢ (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) ∧ (𝑥 ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝐶 ⊆ 𝑥)
11		incom 3949	. . . . 5 ⊢ (𝐷 ∩ ((cls‘𝐽)‘𝑥)) = (((cls‘𝐽)‘𝑥) ∩ 𝐷)
12		simprrr 824	. . . . 5 ⊢ (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) ∧ (𝑥 ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅)
13	11, 12	syl5eq 2807	. . . 4 ⊢ (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) ∧ (𝑥 ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → (𝐷 ∩ ((cls‘𝐽)‘𝑥)) = ∅)
14		simplr2 1263	. . . . 5 ⊢ (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) ∧ (𝑥 ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝐷 ∈ (Clsd‘𝐽))
15	5	cldss 21056	. . . . 5 ⊢ (𝐷 ∈ (Clsd‘𝐽) → 𝐷 ⊆ ∪ 𝐽)
16		reldisj 4164	. . . . 5 ⊢ (𝐷 ⊆ ∪ 𝐽 → ((𝐷 ∩ ((cls‘𝐽)‘𝑥)) = ∅ ↔ 𝐷 ⊆ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))))
17	14, 15, 16	3syl 18	. . . 4 ⊢ (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) ∧ (𝑥 ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → ((𝐷 ∩ ((cls‘𝐽)‘𝑥)) = ∅ ↔ 𝐷 ⊆ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))))
18	13, 17	mpbid 222	. . 3 ⊢ (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) ∧ (𝑥 ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝐷 ⊆ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥)))
19	5	sscls 21083	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ ∪ 𝐽) → 𝑥 ⊆ ((cls‘𝐽)‘𝑥))
20	2, 4, 19	syl2anc 696	. . . . 5 ⊢ (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) ∧ (𝑥 ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝑥 ⊆ ((cls‘𝐽)‘𝑥))
21		ssrin 3982	. . . . 5 ⊢ (𝑥 ⊆ ((cls‘𝐽)‘𝑥) → (𝑥 ∩ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))) ⊆ (((cls‘𝐽)‘𝑥) ∩ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))))
22	20, 21	syl 17	. . . 4 ⊢ (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) ∧ (𝑥 ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → (𝑥 ∩ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))) ⊆ (((cls‘𝐽)‘𝑥) ∩ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))))
23		disjdif 4185	. . . 4 ⊢ (((cls‘𝐽)‘𝑥) ∩ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅
24		sseq0 4119	. . . 4 ⊢ (((𝑥 ∩ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))) ⊆ (((cls‘𝐽)‘𝑥) ∩ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))) ∧ (((cls‘𝐽)‘𝑥) ∩ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅) → (𝑥 ∩ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅)
25	22, 23, 24	sylancl 697	. . 3 ⊢ (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) ∧ (𝑥 ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → (𝑥 ∩ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅)
26		sseq2 3769	. . . . 5 ⊢ (𝑦 = (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥)) → (𝐷 ⊆ 𝑦 ↔ 𝐷 ⊆ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))))
27		ineq2 3952	. . . . . 6 ⊢ (𝑦 = (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥)) → (𝑥 ∩ 𝑦) = (𝑥 ∩ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))))
28	27	eqeq1d 2763	. . . . 5 ⊢ (𝑦 = (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥)) → ((𝑥 ∩ 𝑦) = ∅ ↔ (𝑥 ∩ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅))
29	26, 28	3anbi23d 1551	. . . 4 ⊢ (𝑦 = (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥)) → ((𝐶 ⊆ 𝑥 ∧ 𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅) ↔ (𝐶 ⊆ 𝑥 ∧ 𝐷 ⊆ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥)) ∧ (𝑥 ∩ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅)))
30	29	rspcev 3450	. . 3 ⊢ (((∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥)) ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ 𝐷 ⊆ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥)) ∧ (𝑥 ∩ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅)) → ∃𝑦 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ 𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))
31	9, 10, 18, 25, 30	syl13anc 1479	. 2 ⊢ (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) ∧ (𝑥 ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → ∃𝑦 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ 𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))
32		nrmsep2 21383	. 2 ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → ∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))
33	31, 32	reximddv 3157	1 ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ 𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2140  ∃wrex 3052   ∖ cdif 3713   ∩ cin 3715   ⊆ wss 3716  ∅c0 4059  ∪ cuni 4589  ‘cfv 6050  Topctop 20921  Clsdccld 21043  clsccl 21045  Nrmcnrm 21337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-int 4629  df-iun 4675  df-iin 4676  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-top 20922  df-cld 21046  df-cls 21048  df-nrm 21344

This theorem is referenced by:  isnrm3  21386