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Theorem ntrcls0 22901
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 482 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ 𝐽 ∈ Top)
2 clscld.1 . . . . . 6 𝑋 = βˆͺ 𝐽
32clsss3 22884 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† 𝑋)
42sscls 22881 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ 𝑆 βŠ† ((clsβ€˜π½)β€˜π‘†))
52ntrss 22880 . . . . 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 1370 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π‘†) βŠ† ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘†)))
763adant3 1131 . . 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 4008 . . . 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 1134 . . 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 4398 . 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 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   βŠ† wss 3948  βˆ…c0 4322  βˆͺ cuni 4908  β€˜cfv 6543  Topctop 22716  intcnt 22842  clsccl 22843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  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 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-top 22717  df-cld 22844  df-ntr 22845  df-cls 22846
This theorem is referenced by: (None)
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