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Theorem ntrcls0 22479
Description: A subset whose closure has an empty interior also has an empty interior. (Contributed by NM, 4-Oct-2007.)
Hypothesis
Ref Expression
clscld.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
ntrcls0 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘†)) = βˆ…) β†’ ((intβ€˜π½)β€˜π‘†) = βˆ…)
Proof of Theorem ntrcls0
StepHypRef Expression
1 simpl 483 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ 𝐽 ∈ Top)
2 clscld.1 . . . . . 6 𝑋 = βˆͺ 𝐽
32clsss3 22462 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† 𝑋)
42sscls 22459 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ 𝑆 βŠ† ((clsβ€˜π½)β€˜π‘†))
52ntrss 22458 . . . . 5 ((𝐽 ∈ Top ∧ ((clsβ€˜π½)β€˜π‘†) βŠ† 𝑋 ∧ 𝑆 βŠ† ((clsβ€˜π½)β€˜π‘†)) β†’ ((intβ€˜π½)β€˜π‘†) βŠ† ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘†)))
61, 3, 4, 5syl3anc 1371 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π‘†) βŠ† ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘†)))
763adant3 1132 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘†)) = βˆ…) β†’ ((intβ€˜π½)β€˜π‘†) βŠ† ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘†)))
8 sseq2 3988 . . . 4 (((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘†)) = βˆ… β†’ (((intβ€˜π½)β€˜π‘†) βŠ† ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘†)) ↔ ((intβ€˜π½)β€˜π‘†) βŠ† βˆ…))
983ad2ant3 1135 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘†)) = βˆ…) β†’ (((intβ€˜π½)β€˜π‘†) βŠ† ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘†)) ↔ ((intβ€˜π½)β€˜π‘†) βŠ† βˆ…))
107, 9mpbid 231 . 2 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘†)) = βˆ…) β†’ ((intβ€˜π½)β€˜π‘†) βŠ† βˆ…)
11 ss0 4378 . 2 (((intβ€˜π½)β€˜π‘†) βŠ† βˆ… β†’ ((intβ€˜π½)β€˜π‘†) = βˆ…)
1210, 11syl 17 1 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘†)) = βˆ…) β†’ ((intβ€˜π½)β€˜π‘†) = βˆ…)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   βŠ† wss 3928  βˆ…c0 4302  βˆͺ cuni 4885  β€˜cfv 6516  Topctop 22294  intcnt 22420  clsccl 22421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2702  ax-rep 5262  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404  ax-un 7692
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3365  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-int 4928  df-iun 4976  df-iin 4977  df-br 5126  df-opab 5188  df-mpt 5209  df-id 5551  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-top 22295  df-cld 22422  df-ntr 22423  df-cls 22424
This theorem is referenced by: (None)
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