Theorem nrmsep 23418

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 23397	. . . . . 6 ⊢ (𝐽 ∈ Nrm → 𝐽 ∈ Top)
2	1	ad2antrr 736	. . . . 5 ⊢ (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) ∧ (𝑥 ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝐽 ∈ Top)
3		elssuni 4898	. . . . . 6 ⊢ (𝑥 ∈ 𝐽 → 𝑥 ⊆ ∪ 𝐽)
4	3	ad2antrl 738	. . . . 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 23108	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽))
7	2, 4, 6	syl2anc 593	. . . 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 23092	. . . 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 790	. . 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 791	. . . . 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 1231	. . . . 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 23090	. . . . 5 ⊢ (𝐷 ∈ (Clsd‘𝐽) → 𝐷 ⊆ ∪ 𝐽)
16		reldisj 4408	. . . . 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 234	. . 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 23117	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ ∪ 𝐽) → 𝑥 ⊆ ((cls‘𝐽)‘𝑥))
20	2, 4, 19	syl2anc 593	. . . . 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 4427	. . . 4 ⊢ (((cls‘𝐽)‘𝑥) ∩ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅
23		sseq0 4358	. . . 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 595	. . 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 1461	. . . 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 3582	. . 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 1392	. 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 23417	. 2 ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → ∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))
32	30, 31	reximddv 3179	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 208   ∧ wa 399   ∧ w3a 1099   = wceq 1561   ∈ wcel 2143  ∃wrex 3087   ∖ cdif 3902   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286  ∪ cuni 4866  ‘cfv 6522  Topctop 22954  Clsdccld 23077  clsccl 23079  Nrmcnrm 23371

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-top 22955  df-cld 23080  df-cls 23082  df-nrm 23378

This theorem is referenced by:  isnrm3  23420