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Theorem opnbnd 35210
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 = βˆͺ 𝐽
Assertion
Ref Expression
opnbnd ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ (𝐴 ∈ 𝐽 ↔ (𝐴 ∩ (((clsβ€˜π½)β€˜π΄) ∩ ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝐴)))) = βˆ…))

Proof of Theorem opnbnd
StepHypRef Expression
1 disjdif 4472 . . . . 5 (((intβ€˜π½)β€˜π΄) ∩ (((clsβ€˜π½)β€˜π΄) βˆ– ((intβ€˜π½)β€˜π΄))) = βˆ…
21a1i 11 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ (((intβ€˜π½)β€˜π΄) ∩ (((clsβ€˜π½)β€˜π΄) βˆ– ((intβ€˜π½)β€˜π΄))) = βˆ…)
3 ineq1 4206 . . . . 5 (((intβ€˜π½)β€˜π΄) = 𝐴 β†’ (((intβ€˜π½)β€˜π΄) ∩ (((clsβ€˜π½)β€˜π΄) βˆ– ((intβ€˜π½)β€˜π΄))) = (𝐴 ∩ (((clsβ€˜π½)β€˜π΄) βˆ– ((intβ€˜π½)β€˜π΄))))
43eqeq1d 2735 . . . 4 (((intβ€˜π½)β€˜π΄) = 𝐴 β†’ ((((intβ€˜π½)β€˜π΄) ∩ (((clsβ€˜π½)β€˜π΄) βˆ– ((intβ€˜π½)β€˜π΄))) = βˆ… ↔ (𝐴 ∩ (((clsβ€˜π½)β€˜π΄) βˆ– ((intβ€˜π½)β€˜π΄))) = βˆ…))
52, 4syl5ibcom 244 . . 3 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ (((intβ€˜π½)β€˜π΄) = 𝐴 β†’ (𝐴 ∩ (((clsβ€˜π½)β€˜π΄) βˆ– ((intβ€˜π½)β€˜π΄))) = βˆ…))
6 opnbnd.1 . . . . . . 7 𝑋 = βˆͺ 𝐽
76ntrss2 22561 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π΄) βŠ† 𝐴)
87adantr 482 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) ∧ (𝐴 ∩ (((clsβ€˜π½)β€˜π΄) βˆ– ((intβ€˜π½)β€˜π΄))) = βˆ…) β†’ ((intβ€˜π½)β€˜π΄) βŠ† 𝐴)
9 inssdif0 4370 . . . . . 6 ((𝐴 ∩ ((clsβ€˜π½)β€˜π΄)) βŠ† ((intβ€˜π½)β€˜π΄) ↔ (𝐴 ∩ (((clsβ€˜π½)β€˜π΄) βˆ– ((intβ€˜π½)β€˜π΄))) = βˆ…)
106sscls 22560 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ 𝐴 βŠ† ((clsβ€˜π½)β€˜π΄))
11 df-ss 3966 . . . . . . . . . 10 (𝐴 βŠ† ((clsβ€˜π½)β€˜π΄) ↔ (𝐴 ∩ ((clsβ€˜π½)β€˜π΄)) = 𝐴)
1210, 11sylib 217 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ (𝐴 ∩ ((clsβ€˜π½)β€˜π΄)) = 𝐴)
1312eqcomd 2739 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ 𝐴 = (𝐴 ∩ ((clsβ€˜π½)β€˜π΄)))
14 eqimss 4041 . . . . . . . 8 (𝐴 = (𝐴 ∩ ((clsβ€˜π½)β€˜π΄)) β†’ 𝐴 βŠ† (𝐴 ∩ ((clsβ€˜π½)β€˜π΄)))
1513, 14syl 17 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ 𝐴 βŠ† (𝐴 ∩ ((clsβ€˜π½)β€˜π΄)))
16 sstr 3991 . . . . . . 7 ((𝐴 βŠ† (𝐴 ∩ ((clsβ€˜π½)β€˜π΄)) ∧ (𝐴 ∩ ((clsβ€˜π½)β€˜π΄)) βŠ† ((intβ€˜π½)β€˜π΄)) β†’ 𝐴 βŠ† ((intβ€˜π½)β€˜π΄))
1715, 16sylan 581 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) ∧ (𝐴 ∩ ((clsβ€˜π½)β€˜π΄)) βŠ† ((intβ€˜π½)β€˜π΄)) β†’ 𝐴 βŠ† ((intβ€˜π½)β€˜π΄))
189, 17sylan2br 596 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) ∧ (𝐴 ∩ (((clsβ€˜π½)β€˜π΄) βˆ– ((intβ€˜π½)β€˜π΄))) = βˆ…) β†’ 𝐴 βŠ† ((intβ€˜π½)β€˜π΄))
198, 18eqssd 4000 . . . 4 (((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) ∧ (𝐴 ∩ (((clsβ€˜π½)β€˜π΄) βˆ– ((intβ€˜π½)β€˜π΄))) = βˆ…) β†’ ((intβ€˜π½)β€˜π΄) = 𝐴)
2019ex 414 . . 3 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ ((𝐴 ∩ (((clsβ€˜π½)β€˜π΄) βˆ– ((intβ€˜π½)β€˜π΄))) = βˆ… β†’ ((intβ€˜π½)β€˜π΄) = 𝐴))
215, 20impbid 211 . 2 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ (((intβ€˜π½)β€˜π΄) = 𝐴 ↔ (𝐴 ∩ (((clsβ€˜π½)β€˜π΄) βˆ– ((intβ€˜π½)β€˜π΄))) = βˆ…))
226isopn3 22570 . 2 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ (𝐴 ∈ 𝐽 ↔ ((intβ€˜π½)β€˜π΄) = 𝐴))
236topbnd 35209 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ (((clsβ€˜π½)β€˜π΄) ∩ ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝐴))) = (((clsβ€˜π½)β€˜π΄) βˆ– ((intβ€˜π½)β€˜π΄)))
2423ineq2d 4213 . . 3 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ (𝐴 ∩ (((clsβ€˜π½)β€˜π΄) ∩ ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝐴)))) = (𝐴 ∩ (((clsβ€˜π½)β€˜π΄) βˆ– ((intβ€˜π½)β€˜π΄))))
2524eqeq1d 2735 . 2 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ ((𝐴 ∩ (((clsβ€˜π½)β€˜π΄) ∩ ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝐴)))) = βˆ… ↔ (𝐴 ∩ (((clsβ€˜π½)β€˜π΄) βˆ– ((intβ€˜π½)β€˜π΄))) = βˆ…))
2621, 22, 253bitr4d 311 1 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ (𝐴 ∈ 𝐽 ↔ (𝐴 ∩ (((clsβ€˜π½)β€˜π΄) ∩ ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝐴)))) = βˆ…))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   βˆ– cdif 3946   ∩ cin 3948   βŠ† wss 3949  βˆ…c0 4323  βˆͺ cuni 4909  β€˜cfv 6544  Topctop 22395  intcnt 22521  clsccl 22522
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-top 22396  df-cld 22523  df-ntr 22524  df-cls 22525
This theorem is referenced by:  cldbnd  35211
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