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Theorem opnbnd 36326
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 ∧ 𝐴𝑋) → (𝐴𝐽 ↔ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ∅))

Proof of Theorem opnbnd
StepHypRef Expression
1 disjdif 4472 . . . . 5 (((int‘𝐽)‘𝐴) ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅
21a1i 11 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((int‘𝐽)‘𝐴) ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅)
3 ineq1 4213 . . . . 5 (((int‘𝐽)‘𝐴) = 𝐴 → (((int‘𝐽)‘𝐴) ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))))
43eqeq1d 2739 . . . 4 (((int‘𝐽)‘𝐴) = 𝐴 → ((((int‘𝐽)‘𝐴) ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅ ↔ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅))
52, 4syl5ibcom 245 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((int‘𝐽)‘𝐴) = 𝐴 → (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅))
6 opnbnd.1 . . . . . . 7 𝑋 = 𝐽
76ntrss2 23065 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘𝐴) ⊆ 𝐴)
87adantr 480 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅) → ((int‘𝐽)‘𝐴) ⊆ 𝐴)
9 inssdif0 4374 . . . . . 6 ((𝐴 ∩ ((cls‘𝐽)‘𝐴)) ⊆ ((int‘𝐽)‘𝐴) ↔ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅)
106sscls 23064 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝐴𝑋) → 𝐴 ⊆ ((cls‘𝐽)‘𝐴))
11 dfss2 3969 . . . . . . . . . 10 (𝐴 ⊆ ((cls‘𝐽)‘𝐴) ↔ (𝐴 ∩ ((cls‘𝐽)‘𝐴)) = 𝐴)
1210, 11sylib 218 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∩ ((cls‘𝐽)‘𝐴)) = 𝐴)
1312eqcomd 2743 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐴𝑋) → 𝐴 = (𝐴 ∩ ((cls‘𝐽)‘𝐴)))
14 eqimss 4042 . . . . . . . 8 (𝐴 = (𝐴 ∩ ((cls‘𝐽)‘𝐴)) → 𝐴 ⊆ (𝐴 ∩ ((cls‘𝐽)‘𝐴)))
1513, 14syl 17 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐴𝑋) → 𝐴 ⊆ (𝐴 ∩ ((cls‘𝐽)‘𝐴)))
16 sstr 3992 . . . . . . 7 ((𝐴 ⊆ (𝐴 ∩ ((cls‘𝐽)‘𝐴)) ∧ (𝐴 ∩ ((cls‘𝐽)‘𝐴)) ⊆ ((int‘𝐽)‘𝐴)) → 𝐴 ⊆ ((int‘𝐽)‘𝐴))
1715, 16sylan 580 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝐴 ∩ ((cls‘𝐽)‘𝐴)) ⊆ ((int‘𝐽)‘𝐴)) → 𝐴 ⊆ ((int‘𝐽)‘𝐴))
189, 17sylan2br 595 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅) → 𝐴 ⊆ ((int‘𝐽)‘𝐴))
198, 18eqssd 4001 . . . 4 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅) → ((int‘𝐽)‘𝐴) = 𝐴)
2019ex 412 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅ → ((int‘𝐽)‘𝐴) = 𝐴))
215, 20impbid 212 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((int‘𝐽)‘𝐴) = 𝐴 ↔ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅))
226isopn3 23074 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴𝐽 ↔ ((int‘𝐽)‘𝐴) = 𝐴))
236topbnd 36325 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) = (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴)))
2423ineq2d 4220 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))))
2524eqeq1d 2739 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝐴 ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ∅ ↔ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅))
2621, 22, 253bitr4d 311 1 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴𝐽 ↔ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  cdif 3948  cin 3950  wss 3951  c0 4333   cuni 4907  cfv 6561  Topctop 22899  intcnt 23025  clsccl 23026
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-top 22900  df-cld 23027  df-ntr 23028  df-cls 23029
This theorem is referenced by:  cldbnd  36327
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