ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ntrcls0 GIF version
Theorem ntrcls0 13771
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 109 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ 𝐽 ∈ Top)
2 clscld.1 . . . . . 6 𝑋 = βˆͺ 𝐽
32clsss3 13770 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† 𝑋)
42sscls 13760 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ 𝑆 βŠ† ((clsβ€˜π½)β€˜π‘†))
52ntrss 13759 . . . . 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 1238 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π‘†) βŠ† ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘†)))
763adant3 1017 . . 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 3181 . . . 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 1020 . . 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 147 . 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 3465 . 2 (((intβ€˜π½)β€˜π‘†) βŠ† βˆ… β†’ ((intβ€˜π½)β€˜π‘†) = βˆ…)
1210, 11syl 14 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 set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978   = wceq 1353   ∈ wcel 2148   βŠ† wss 3131  βˆ…c0 3424  βˆͺ cuni 3811  β€˜cfv 5218  Topctop 13637  intcnt 13733  clsccl 13734
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-top 13638  df-cld 13735  df-ntr 13736  df-cls 13737
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator