Theorem regsep2 23385

Description: In a regular space, a closed set is separated by open sets from a point not in it. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 25-Aug-2015.)

Hypothesis

Ref	Expression
t1sep.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
regsep2	⊢ ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ 𝐴 ∈ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐶,𝑦   𝑥,𝐽,𝑦   𝑥,𝑋,𝑦

Proof of Theorem regsep2

Step	Hyp	Ref	Expression
1		regtop 23342	. . . . . . 7 ⊢ (𝐽 ∈ Reg → 𝐽 ∈ Top)
2	1	ad2antrr 726	. . . . . 6 ⊢ (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) ∧ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))) → 𝐽 ∈ Top)
3		elssuni 4936	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐽 → 𝑦 ⊆ ∪ 𝐽)
4		t1sep.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
5	3, 4	sseqtrrdi 4024	. . . . . . 7 ⊢ (𝑦 ∈ 𝐽 → 𝑦 ⊆ 𝑋)
6	5	ad2antrl 728	. . . . . 6 ⊢ (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) ∧ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))) → 𝑦 ⊆ 𝑋)
7	4	clscld 23056	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝑋) → ((cls‘𝐽)‘𝑦) ∈ (Clsd‘𝐽))
8	2, 6, 7	syl2anc 584	. . . . 5 ⊢ (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) ∧ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))) → ((cls‘𝐽)‘𝑦) ∈ (Clsd‘𝐽))
9	4	cldopn 23040	. . . . 5 ⊢ (((cls‘𝐽)‘𝑦) ∈ (Clsd‘𝐽) → (𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∈ 𝐽)
10	8, 9	syl 17	. . . 4 ⊢ (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) ∧ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))) → (𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∈ 𝐽)
11		simprrr 781	. . . . 5 ⊢ (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) ∧ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))) → ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶))
12	4	clsss3 23068	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝑋) → ((cls‘𝐽)‘𝑦) ⊆ 𝑋)
13	2, 6, 12	syl2anc 584	. . . . . 6 ⊢ (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) ∧ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))) → ((cls‘𝐽)‘𝑦) ⊆ 𝑋)
14		simplr1 1215	. . . . . . 7 ⊢ (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) ∧ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))) → 𝐶 ∈ (Clsd‘𝐽))
15	4	cldss 23038	. . . . . . 7 ⊢ (𝐶 ∈ (Clsd‘𝐽) → 𝐶 ⊆ 𝑋)
16	14, 15	syl 17	. . . . . 6 ⊢ (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) ∧ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))) → 𝐶 ⊆ 𝑋)
17		ssconb 4141	. . . . . 6 ⊢ ((((cls‘𝐽)‘𝑦) ⊆ 𝑋 ∧ 𝐶 ⊆ 𝑋) → (((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶) ↔ 𝐶 ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝑦))))
18	13, 16, 17	syl2anc 584	. . . . 5 ⊢ (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) ∧ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))) → (((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶) ↔ 𝐶 ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝑦))))
19	11, 18	mpbid 232	. . . 4 ⊢ (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) ∧ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))) → 𝐶 ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝑦)))
20		simprrl 780	. . . 4 ⊢ (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) ∧ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))) → 𝐴 ∈ 𝑦)
21	4	sscls 23065	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝑋) → 𝑦 ⊆ ((cls‘𝐽)‘𝑦))
22	2, 6, 21	syl2anc 584	. . . . . 6 ⊢ (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) ∧ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))) → 𝑦 ⊆ ((cls‘𝐽)‘𝑦))
23		sslin 4242	. . . . . 6 ⊢ (𝑦 ⊆ ((cls‘𝐽)‘𝑦) → ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ 𝑦) ⊆ ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ ((cls‘𝐽)‘𝑦)))
24	22, 23	syl 17	. . . . 5 ⊢ (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) ∧ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))) → ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ 𝑦) ⊆ ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ ((cls‘𝐽)‘𝑦)))
25		disjdifr 4472	. . . . 5 ⊢ ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ ((cls‘𝐽)‘𝑦)) = ∅
26		sseq0 4402	. . . . 5 ⊢ ((((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ 𝑦) ⊆ ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ ((cls‘𝐽)‘𝑦)) ∧ ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ ((cls‘𝐽)‘𝑦)) = ∅) → ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ 𝑦) = ∅)
27	24, 25, 26	sylancl 586	. . . 4 ⊢ (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) ∧ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))) → ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ 𝑦) = ∅)
28		sseq2 4009	. . . . . 6 ⊢ (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑦)) → (𝐶 ⊆ 𝑥 ↔ 𝐶 ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝑦))))
29		ineq1 4212	. . . . . . 7 ⊢ (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑦)) → (𝑥 ∩ 𝑦) = ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ 𝑦))
30	29	eqeq1d 2738	. . . . . 6 ⊢ (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑦)) → ((𝑥 ∩ 𝑦) = ∅ ↔ ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ 𝑦) = ∅))
31	28, 30	3anbi13d 1439	. . . . 5 ⊢ (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑦)) → ((𝐶 ⊆ 𝑥 ∧ 𝐴 ∈ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅) ↔ (𝐶 ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∧ 𝐴 ∈ 𝑦 ∧ ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ 𝑦) = ∅)))
32	31	rspcev 3621	. . . 4 ⊢ (((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∈ 𝐽 ∧ (𝐶 ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∧ 𝐴 ∈ 𝑦 ∧ ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ 𝑦) = ∅)) → ∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ 𝐴 ∈ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))
33	10, 19, 20, 27, 32	syl13anc 1373	. . 3 ⊢ (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) ∧ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))) → ∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ 𝐴 ∈ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))
34		simpl 482	. . . 4 ⊢ ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) → 𝐽 ∈ Reg)
35		simpr1 1194	. . . . 5 ⊢ ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) → 𝐶 ∈ (Clsd‘𝐽))
36	4	cldopn 23040	. . . . 5 ⊢ (𝐶 ∈ (Clsd‘𝐽) → (𝑋 ∖ 𝐶) ∈ 𝐽)
37	35, 36	syl 17	. . . 4 ⊢ ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) → (𝑋 ∖ 𝐶) ∈ 𝐽)
38		simpr2 1195	. . . . 5 ⊢ ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) → 𝐴 ∈ 𝑋)
39		simpr3 1196	. . . . 5 ⊢ ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) → ¬ 𝐴 ∈ 𝐶)
40	38, 39	eldifd 3961	. . . 4 ⊢ ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) → 𝐴 ∈ (𝑋 ∖ 𝐶))
41		regsep 23343	. . . 4 ⊢ ((𝐽 ∈ Reg ∧ (𝑋 ∖ 𝐶) ∈ 𝐽 ∧ 𝐴 ∈ (𝑋 ∖ 𝐶)) → ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))
42	34, 37, 40, 41	syl3anc 1372	. . 3 ⊢ ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) → ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))
43	33, 42	reximddv 3170	. 2 ⊢ ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) → ∃𝑦 ∈ 𝐽 ∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ 𝐴 ∈ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))
44		rexcom 3289	. 2 ⊢ (∃𝑦 ∈ 𝐽 ∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ 𝐴 ∈ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅) ↔ ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ 𝐴 ∈ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))
45	43, 44	sylib 218	1 ⊢ ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ 𝐴 ∈ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ∃wrex 3069   ∖ cdif 3947   ∩ cin 3949   ⊆ wss 3950  ∅c0 4332  ∪ cuni 4906  ‘cfv 6560  Topctop 22900  Clsdccld 23025  clsccl 23027  Regcreg 23318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-top 22901  df-cld 23028  df-cls 23030  df-reg 23325

This theorem is referenced by:  isreg2  23386