Theorem nrmsep 23514

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 23493	. . . . . 6 ⊢ (𝐽 ∈ Nrm → 𝐽 ∈ Top)
2	1	ad2antrr 738	. . . . 5 ⊢ (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) ∧ (𝑥 ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝐽 ∈ Top)
3		elssuni 4904	. . . . . 6 ⊢ (𝑥 ∈ 𝐽 → 𝑥 ⊆ ∪ 𝐽)
4	3	ad2antrl 740	. . . . 5 ⊢ (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) ∧ (𝑥 ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝑥 ⊆ ∪ 𝐽)
5		eqid 2763	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
6	5	clscld 23204	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽))
7	2, 4, 6	syl2anc 595	. . . 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 23188	. . . 4 ⊢ (((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥)) ∈ 𝐽)
9	7, 8	syl 18	. . 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 792	. . 3 ⊢ (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) ∧ (𝑥 ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝐶 ⊆ 𝑥)
11		incom 4162	. . . . 5 ⊢ (𝐷 ∩ ((cls‘𝐽)‘𝑥)) = (((cls‘𝐽)‘𝑥) ∩ 𝐷)
12		simprrr 793	. . . . 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	eqtrid 2810	. . . 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 1235	. . . . 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 23186	. . . . 5 ⊢ (𝐷 ∈ (Clsd‘𝐽) → 𝐷 ⊆ ∪ 𝐽)
16		reldisj 4413	. . . . 5 ⊢ (𝐷 ⊆ ∪ 𝐽 → ((𝐷 ∩ ((cls‘𝐽)‘𝑥)) = ∅ ↔ 𝐷 ⊆ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))))
17	14, 15, 16	3syl 19	. . . 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 235	. . 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 23213	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ ∪ 𝐽) → 𝑥 ⊆ ((cls‘𝐽)‘𝑥))
20	2, 4, 19	syl2anc 595	. . . . 5 ⊢ (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) ∧ (𝑥 ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝑥 ⊆ ((cls‘𝐽)‘𝑥))
21	20	ssrind 4196	. . . 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))))
22		disjdif 4433	. . . 4 ⊢ (((cls‘𝐽)‘𝑥) ∩ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅
23		sseq0 4361	. . . 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))) = ∅)
24	21, 22, 23	sylancl 597	. . 3 ⊢ (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) ∧ (𝑥 ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → (𝑥 ∩ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅)
25		sseq2 3963	. . . . 5 ⊢ (𝑦 = (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥)) → (𝐷 ⊆ 𝑦 ↔ 𝐷 ⊆ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))))
26		ineq2 4167	. . . . . 6 ⊢ (𝑦 = (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥)) → (𝑥 ∩ 𝑦) = (𝑥 ∩ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))))
27	26	eqeq1d 2765	. . . . 5 ⊢ (𝑦 = (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥)) → ((𝑥 ∩ 𝑦) = ∅ ↔ (𝑥 ∩ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅))
28	25, 27	3anbi23d 1467	. . . 4 ⊢ (𝑦 = (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥)) → ((𝐶 ⊆ 𝑥 ∧ 𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅) ↔ (𝐶 ⊆ 𝑥 ∧ 𝐷 ⊆ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥)) ∧ (𝑥 ∩ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅)))
29	28	rspcev 3581	. . 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) = ∅))
30	9, 10, 18, 24, 29	syl13anc 1399	. 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) = ∅))
31		nrmsep2 23513	. 2 ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → ∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))
32	30, 31	reximddv 3181	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 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∃wrex 3089   ∖ cdif 3902   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286  ∪ cuni 4872  ‘cfv 6536  Topctop 23050  Clsdccld 23173  clsccl 23175  Nrmcnrm 23467

This theorem was proved from 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-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem 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-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  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-int 4913  df-iun 4958  df-iin 4959  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-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-top 23051  df-cld 23176  df-cls 23178  df-nrm 23474

This theorem is referenced by:  isnrm3  23516