Theorem opnbnd 36286

Description: A set is open iff it is disjoint from its boundary. (Contributed by Jeff Hankins, 23-Sep-2009.)

Hypothesis

Ref	Expression
opnbnd.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
opnbnd	⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ 𝐽 ↔ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))) = ∅))

Proof of Theorem opnbnd

Step	Hyp	Ref	Expression
1		disjdif 4431	. . . . 5 ⊢ (((int‘𝐽)‘𝐴) ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅
2	1	a1i 11	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((int‘𝐽)‘𝐴) ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅)
3		ineq1 4172	. . . . 5 ⊢ (((int‘𝐽)‘𝐴) = 𝐴 → (((int‘𝐽)‘𝐴) ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))))
4	3	eqeq1d 2731	. . . 4 ⊢ (((int‘𝐽)‘𝐴) = 𝐴 → ((((int‘𝐽)‘𝐴) ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅ ↔ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅))
5	2, 4	syl5ibcom 245	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((int‘𝐽)‘𝐴) = 𝐴 → (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅))
6		opnbnd.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
7	6	ntrss2 22920	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐽)‘𝐴) ⊆ 𝐴)
8	7	adantr 480	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅) → ((int‘𝐽)‘𝐴) ⊆ 𝐴)
9		inssdif0 4333	. . . . . 6 ⊢ ((𝐴 ∩ ((cls‘𝐽)‘𝐴)) ⊆ ((int‘𝐽)‘𝐴) ↔ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅)
10	6	sscls 22919	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ ((cls‘𝐽)‘𝐴))
11		dfss2 3929	. . . . . . . . . 10 ⊢ (𝐴 ⊆ ((cls‘𝐽)‘𝐴) ↔ (𝐴 ∩ ((cls‘𝐽)‘𝐴)) = 𝐴)
12	10, 11	sylib 218	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∩ ((cls‘𝐽)‘𝐴)) = 𝐴)
13	12	eqcomd 2735	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → 𝐴 = (𝐴 ∩ ((cls‘𝐽)‘𝐴)))
14		eqimss 4002	. . . . . . . 8 ⊢ (𝐴 = (𝐴 ∩ ((cls‘𝐽)‘𝐴)) → 𝐴 ⊆ (𝐴 ∩ ((cls‘𝐽)‘𝐴)))
15	13, 14	syl 17	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ (𝐴 ∩ ((cls‘𝐽)‘𝐴)))
16		sstr 3952	. . . . . . 7 ⊢ ((𝐴 ⊆ (𝐴 ∩ ((cls‘𝐽)‘𝐴)) ∧ (𝐴 ∩ ((cls‘𝐽)‘𝐴)) ⊆ ((int‘𝐽)‘𝐴)) → 𝐴 ⊆ ((int‘𝐽)‘𝐴))
17	15, 16	sylan 580	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝐴 ∩ ((cls‘𝐽)‘𝐴)) ⊆ ((int‘𝐽)‘𝐴)) → 𝐴 ⊆ ((int‘𝐽)‘𝐴))
18	9, 17	sylan2br 595	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅) → 𝐴 ⊆ ((int‘𝐽)‘𝐴))
19	8, 18	eqssd 3961	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅) → ((int‘𝐽)‘𝐴) = 𝐴)
20	19	ex 412	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅ → ((int‘𝐽)‘𝐴) = 𝐴))
21	5, 20	impbid 212	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((int‘𝐽)‘𝐴) = 𝐴 ↔ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅))
22	6	isopn3 22929	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ 𝐽 ↔ ((int‘𝐽)‘𝐴) = 𝐴))
23	6	topbnd 36285	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) = (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴)))
24	23	ineq2d 4179	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))) = (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))))
25	24	eqeq1d 2731	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((𝐴 ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))) = ∅ ↔ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅))
26	21, 22, 25	3bitr4d 311	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ 𝐽 ↔ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))) = ∅))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∖ cdif 3908   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292  ∪ cuni 4867  ‘cfv 6499  Topctop 22756  intcnt 22880  clsccl 22881

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 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-top 22757  df-cld 22882  df-ntr 22883  df-cls 22884

This theorem is referenced by:  cldbnd  36287