Theorem opnbnd 34441

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 4402	. . . . 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 4136	. . . . 5 ⊢ (((int‘𝐽)‘𝐴) = 𝐴 → (((int‘𝐽)‘𝐴) ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))))
4	3	eqeq1d 2740	. . . 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 244	. . 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 22116	. . . . . 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 4300	. . . . . 6 ⊢ ((𝐴 ∩ ((cls‘𝐽)‘𝐴)) ⊆ ((int‘𝐽)‘𝐴) ↔ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅)
10	6	sscls 22115	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ ((cls‘𝐽)‘𝐴))
11		df-ss 3900	. . . . . . . . . 10 ⊢ (𝐴 ⊆ ((cls‘𝐽)‘𝐴) ↔ (𝐴 ∩ ((cls‘𝐽)‘𝐴)) = 𝐴)
12	10, 11	sylib 217	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∩ ((cls‘𝐽)‘𝐴)) = 𝐴)
13	12	eqcomd 2744	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → 𝐴 = (𝐴 ∩ ((cls‘𝐽)‘𝐴)))
14		eqimss 3973	. . . . . . . 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 3925	. . . . . . 7 ⊢ ((𝐴 ⊆ (𝐴 ∩ ((cls‘𝐽)‘𝐴)) ∧ (𝐴 ∩ ((cls‘𝐽)‘𝐴)) ⊆ ((int‘𝐽)‘𝐴)) → 𝐴 ⊆ ((int‘𝐽)‘𝐴))
17	15, 16	sylan 579	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝐴 ∩ ((cls‘𝐽)‘𝐴)) ⊆ ((int‘𝐽)‘𝐴)) → 𝐴 ⊆ ((int‘𝐽)‘𝐴))
18	9, 17	sylan2br 594	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅) → 𝐴 ⊆ ((int‘𝐽)‘𝐴))
19	8, 18	eqssd 3934	. . . 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 211	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((int‘𝐽)‘𝐴) = 𝐴 ↔ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅))
22	6	isopn3 22125	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ 𝐽 ↔ ((int‘𝐽)‘𝐴) = 𝐴))
23	6	topbnd 34440	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) = (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴)))
24	23	ineq2d 4143	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))) = (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))))
25	24	eqeq1d 2740	. 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 310	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ 𝐽 ↔ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))) = ∅))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ∖ cdif 3880   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  ∪ cuni 4836  ‘cfv 6418  Topctop 21950  intcnt 22076  clsccl 22077

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-top 21951  df-cld 22078  df-ntr 22079  df-cls 22080

This theorem is referenced by:  cldbnd  34442