Theorem nrmsep 23316

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 23295	. . . . . 6 ⊢ (𝐽 ∈ Nrm → 𝐽 ∈ Top)
2	1	ad2antrr 727	. . . . 5 ⊢ (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) ∧ (𝑥 ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝐽 ∈ Top)
3		elssuni 4896	. . . . . 6 ⊢ (𝑥 ∈ 𝐽 → 𝑥 ⊆ ∪ 𝐽)
4	3	ad2antrl 729	. . . . 5 ⊢ (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) ∧ (𝑥 ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝑥 ⊆ ∪ 𝐽)
5		eqid 2737	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
6	5	clscld 23006	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽))
7	2, 4, 6	syl2anc 585	. . . 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 22990	. . . 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 781	. . 3 ⊢ (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) ∧ (𝑥 ∈ 𝐽 ∧ (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝐶 ⊆ 𝑥)
11		incom 4163	. . . . 5 ⊢ (𝐷 ∩ ((cls‘𝐽)‘𝑥)) = (((cls‘𝐽)‘𝑥) ∩ 𝐷)
12		simprrr 782	. . . . 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 2784	. . . 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 1218	. . . . 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 22988	. . . . 5 ⊢ (𝐷 ∈ (Clsd‘𝐽) → 𝐷 ⊆ ∪ 𝐽)
16		reldisj 4407	. . . . 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 232	. . 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 23015	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ ∪ 𝐽) → 𝑥 ⊆ ((cls‘𝐽)‘𝑥))
20	2, 4, 19	syl2anc 585	. . . . 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 4198	. . . 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 4426	. . . 4 ⊢ (((cls‘𝐽)‘𝑥) ∩ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅
23		sseq0 4357	. . . 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 587	. . 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 3962	. . . . 5 ⊢ (𝑦 = (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥)) → (𝐷 ⊆ 𝑦 ↔ 𝐷 ⊆ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))))
26		ineq2 4168	. . . . . 6 ⊢ (𝑦 = (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥)) → (𝑥 ∩ 𝑦) = (𝑥 ∩ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))))
27	26	eqeq1d 2739	. . . . 5 ⊢ (𝑦 = (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥)) → ((𝑥 ∩ 𝑦) = ∅ ↔ (𝑥 ∩ (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅))
28	25, 27	3anbi23d 1442	. . . 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 3578	. . 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 1375	. 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 23315	. 2 ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → ∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))
32	30, 31	reximddv 3154	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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3062   ∖ cdif 3900   ∩ cin 3902   ⊆ wss 3903  ∅c0 4287  ∪ cuni 4865  ‘cfv 6500  Topctop 22852  Clsdccld 22975  clsccl 22977  Nrmcnrm 23269

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 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-top 22853  df-cld 22978  df-cls 22980  df-nrm 23276

This theorem is referenced by:  isnrm3  23318