Theorem nrmsep2 23298

Description: In a normal space, any two disjoint closed sets have the property that each one is a subset of an open set whose closure is disjoint from the other. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 24-Aug-2015.)

Assertion

Ref	Expression
nrmsep2	⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → ∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))

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

Proof of Theorem nrmsep2

Step	Hyp	Ref	Expression
1		simpl 482	. . 3 ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → 𝐽 ∈ Nrm)
2		simpr2 1196	. . . 4 ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → 𝐷 ∈ (Clsd‘𝐽))
3		eqid 2734	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
4	3	cldopn 22973	. . . 4 ⊢ (𝐷 ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ 𝐷) ∈ 𝐽)
5	2, 4	syl 17	. . 3 ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → (∪ 𝐽 ∖ 𝐷) ∈ 𝐽)
6		simpr1 1195	. . 3 ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → 𝐶 ∈ (Clsd‘𝐽))
7		simpr3 1197	. . . 4 ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → (𝐶 ∩ 𝐷) = ∅)
8	3	cldss 22971	. . . . 5 ⊢ (𝐶 ∈ (Clsd‘𝐽) → 𝐶 ⊆ ∪ 𝐽)
9		reldisj 4403	. . . . 5 ⊢ (𝐶 ⊆ ∪ 𝐽 → ((𝐶 ∩ 𝐷) = ∅ ↔ 𝐶 ⊆ (∪ 𝐽 ∖ 𝐷)))
10	6, 8, 9	3syl 18	. . . 4 ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → ((𝐶 ∩ 𝐷) = ∅ ↔ 𝐶 ⊆ (∪ 𝐽 ∖ 𝐷)))
11	7, 10	mpbid 232	. . 3 ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → 𝐶 ⊆ (∪ 𝐽 ∖ 𝐷))
12		nrmsep3 23297	. . 3 ⊢ ((𝐽 ∈ Nrm ∧ ((∪ 𝐽 ∖ 𝐷) ∈ 𝐽 ∧ 𝐶 ∈ (Clsd‘𝐽) ∧ 𝐶 ⊆ (∪ 𝐽 ∖ 𝐷))) → ∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ (∪ 𝐽 ∖ 𝐷)))
13	1, 5, 6, 11, 12	syl13anc 1374	. 2 ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → ∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ (∪ 𝐽 ∖ 𝐷)))
14		ssdifin0 4436	. . . 4 ⊢ (((cls‘𝐽)‘𝑥) ⊆ (∪ 𝐽 ∖ 𝐷) → (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅)
15	14	anim2i 617	. . 3 ⊢ ((𝐶 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ (∪ 𝐽 ∖ 𝐷)) → (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))
16	15	reximi 3072	. 2 ⊢ (∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ (∪ 𝐽 ∖ 𝐷)) → ∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))
17	13, 16	syl 17	1 ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → ∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∃wrex 3058   ∖ cdif 3896   ∩ cin 3898   ⊆ wss 3899  ∅c0 4283  ∪ cuni 4861  ‘cfv 6490  Clsdccld 22958  clsccl 22960  Nrmcnrm 23252

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-iota 6446  df-fun 6492  df-fn 6493  df-fv 6498  df-top 22836  df-cld 22961  df-nrm 23259

This theorem is referenced by:  nrmsep  23299  isnrm2  23300