Theorem regsep2 23319

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 23276	. . . . . . 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 4918	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐽 → 𝑦 ⊆ ∪ 𝐽)
4		t1sep.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
5	3, 4	sseqtrrdi 4005	. . . . . . 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 22990	. . . . . 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 22974	. . . . 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 23002	. . . . . . 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 1216	. . . . . . 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 22972	. . . . . . 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 4122	. . . . . 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 22999	. . . . . . 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 4223	. . . . . 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 4453	. . . . 5 ⊢ ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ ((cls‘𝐽)‘𝑦)) = ∅
26		sseq0 4383	. . . . 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 3990	. . . . . 6 ⊢ (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑦)) → (𝐶 ⊆ 𝑥 ↔ 𝐶 ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝑦))))
29		ineq1 4193	. . . . . . 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 1440	. . . . 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 3606	. . . 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 1374	. . 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 1195	. . . . 5 ⊢ ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) → 𝐶 ∈ (Clsd‘𝐽))
36	4	cldopn 22974	. . . . 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 1196	. . . . 5 ⊢ ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) → 𝐴 ∈ 𝑋)
39		simpr3 1197	. . . . 5 ⊢ ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) → ¬ 𝐴 ∈ 𝐶)
40	38, 39	eldifd 3942	. . . 4 ⊢ ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) → 𝐴 ∈ (𝑋 ∖ 𝐶))
41		regsep 23277	. . . 4 ⊢ ((𝐽 ∈ Reg ∧ (𝑋 ∖ 𝐶) ∈ 𝐽 ∧ 𝐴 ∈ (𝑋 ∖ 𝐶)) → ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))
42	34, 37, 40, 41	syl3anc 1373	. . 3 ⊢ ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) → ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))
43	33, 42	reximddv 3157	. 2 ⊢ ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) → ∃𝑦 ∈ 𝐽 ∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ 𝐴 ∈ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))
44		rexcom 3275	. 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 1540   ∈ wcel 2109  ∃wrex 3061   ∖ cdif 3928   ∩ cin 3930   ⊆ wss 3931  ∅c0 4313  ∪ cuni 4888  ‘cfv 6536  Topctop 22836  Clsdccld 22959  clsccl 22961  Regcreg 23252

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-iin 4975  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-top 22837  df-cld 22962  df-cls 22964  df-reg 23259

This theorem is referenced by:  isreg2  23320