Theorem regsep2 22527

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 22484	. . . . . . 7 ⊢ (𝐽 ∈ Reg → 𝐽 ∈ Top)
2	1	ad2antrr 723	. . . . . 6 ⊢ (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) ∧ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))) → 𝐽 ∈ Top)
3		elssuni 4871	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐽 → 𝑦 ⊆ ∪ 𝐽)
4		t1sep.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
5	3, 4	sseqtrrdi 3972	. . . . . . 7 ⊢ (𝑦 ∈ 𝐽 → 𝑦 ⊆ 𝑋)
6	5	ad2antrl 725	. . . . . 6 ⊢ (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) ∧ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))) → 𝑦 ⊆ 𝑋)
7	4	clscld 22198	. . . . . 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 22182	. . . . 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 779	. . . . 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 22210	. . . . . . 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 1214	. . . . . . 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 22180	. . . . . . 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 4072	. . . . . 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 231	. . . 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 778	. . . 4 ⊢ (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) ∧ (𝑦 ∈ 𝐽 ∧ (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))) → 𝐴 ∈ 𝑦)
21	4	sscls 22207	. . . . . . 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 4168	. . . . . 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 4406	. . . . 5 ⊢ ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ ((cls‘𝐽)‘𝑦)) = ∅
26		sseq0 4333	. . . . 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 3947	. . . . . 6 ⊢ (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑦)) → (𝐶 ⊆ 𝑥 ↔ 𝐶 ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝑦))))
29		ineq1 4139	. . . . . . 7 ⊢ (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑦)) → (𝑥 ∩ 𝑦) = ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ 𝑦))
30	29	eqeq1d 2740	. . . . . 6 ⊢ (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑦)) → ((𝑥 ∩ 𝑦) = ∅ ↔ ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ 𝑦) = ∅))
31	28, 30	3anbi13d 1437	. . . . 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 3561	. . . 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 1371	. . 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 483	. . . 4 ⊢ ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) → 𝐽 ∈ Reg)
35		simpr1 1193	. . . . 5 ⊢ ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) → 𝐶 ∈ (Clsd‘𝐽))
36	4	cldopn 22182	. . . . 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 1194	. . . . 5 ⊢ ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) → 𝐴 ∈ 𝑋)
39		simpr3 1195	. . . . 5 ⊢ ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) → ¬ 𝐴 ∈ 𝐶)
40	38, 39	eldifd 3898	. . . 4 ⊢ ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) → 𝐴 ∈ (𝑋 ∖ 𝐶))
41		regsep 22485	. . . 4 ⊢ ((𝐽 ∈ Reg ∧ (𝑋 ∖ 𝐶) ∈ 𝐽 ∧ 𝐴 ∈ (𝑋 ∖ 𝐶)) → ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))
42	34, 37, 40, 41	syl3anc 1370	. . 3 ⊢ ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) → ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋 ∖ 𝐶)))
43	33, 42	reximddv 3204	. 2 ⊢ ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) → ∃𝑦 ∈ 𝐽 ∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ 𝐴 ∈ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))
44		rexcom 3234	. 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 217	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 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∃wrex 3065   ∖ cdif 3884   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  ∪ cuni 4839  ‘cfv 6433  Topctop 22042  Clsdccld 22167  clsccl 22169  Regcreg 22460

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-top 22043  df-cld 22170  df-cls 22172  df-reg 22467

This theorem is referenced by:  isreg2  22528