Theorem nrmsep2 23321

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 1197	. . . 4 ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → 𝐷 ∈ (Clsd‘𝐽))
3		eqid 2736	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
4	3	cldopn 22996	. . . 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 1196	. . 3 ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → 𝐶 ∈ (Clsd‘𝐽))
7		simpr3 1198	. . . 4 ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → (𝐶 ∩ 𝐷) = ∅)
8	3	cldss 22994	. . . . 5 ⊢ (𝐶 ∈ (Clsd‘𝐽) → 𝐶 ⊆ ∪ 𝐽)
9		reldisj 4393	. . . . 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 23320	. . 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 1375	. 2 ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → ∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ (∪ 𝐽 ∖ 𝐷)))
14		ssdifin0 4425	. . . 4 ⊢ (((cls‘𝐽)‘𝑥) ⊆ (∪ 𝐽 ∖ 𝐷) → (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅)
15	14	anim2i 618	. . 3 ⊢ ((𝐶 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ (∪ 𝐽 ∖ 𝐷)) → (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))
16	15	reximi 3075	. 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 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3061   ∖ cdif 3886   ∩ cin 3888   ⊆ wss 3889  ∅c0 4273  ∪ cuni 4850  ‘cfv 6498  Clsdccld 22981  clsccl 22983  Nrmcnrm 23275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6454  df-fun 6500  df-fn 6501  df-fv 6506  df-top 22859  df-cld 22984  df-nrm 23282

This theorem is referenced by:  nrmsep  23322  isnrm2  23323