Theorem nrmsep2 23522

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 487	. . 3 ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → 𝐽 ∈ Nrm)
2		simpr2 1214	. . . 4 ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → 𝐷 ∈ (Clsd‘𝐽))
3		eqid 2763	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
4	3	cldopn 23197	. . . 4 ⊢ (𝐷 ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ 𝐷) ∈ 𝐽)
5	2, 4	syl 18	. . 3 ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → (∪ 𝐽 ∖ 𝐷) ∈ 𝐽)
6		simpr1 1213	. . 3 ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → 𝐶 ∈ (Clsd‘𝐽))
7		simpr3 1215	. . . 4 ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → (𝐶 ∩ 𝐷) = ∅)
8	3	cldss 23195	. . . . 5 ⊢ (𝐶 ∈ (Clsd‘𝐽) → 𝐶 ⊆ ∪ 𝐽)
9		reldisj 4413	. . . . 5 ⊢ (𝐶 ⊆ ∪ 𝐽 → ((𝐶 ∩ 𝐷) = ∅ ↔ 𝐶 ⊆ (∪ 𝐽 ∖ 𝐷)))
10	6, 8, 9	3syl 19	. . . 4 ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → ((𝐶 ∩ 𝐷) = ∅ ↔ 𝐶 ⊆ (∪ 𝐽 ∖ 𝐷)))
11	7, 10	mpbid 235	. . 3 ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → 𝐶 ⊆ (∪ 𝐽 ∖ 𝐷))
12		nrmsep3 23521	. . 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 1399	. 2 ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → ∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ (∪ 𝐽 ∖ 𝐷)))
14		ssdifin0 4446	. . . 4 ⊢ (((cls‘𝐽)‘𝑥) ⊆ (∪ 𝐽 ∖ 𝐷) → (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅)
15	14	anim2i 628	. . 3 ⊢ ((𝐶 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ (∪ 𝐽 ∖ 𝐷)) → (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))
16	15	reximi 3103	. 2 ⊢ (∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ (∪ 𝐽 ∖ 𝐷)) → ∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))
17	13, 16	syl 18	1 ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → ∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∃wrex 3089   ∖ cdif 3902   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286  ∪ cuni 4872  ‘cfv 6536  Clsdccld 23182  clsccl 23184  Nrmcnrm 23476

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fn 6539  df-fv 6544  df-top 23060  df-cld 23185  df-nrm 23483

This theorem is used by:  nrmsep  23523  isnrm2  23524