Theorem ntrcls0 8983

Description: A subset whose closure has an empty interior also has an empty interior.

Hypothesis

Ref	Expression
clscld.1	|- X = U.J

Assertion

Ref	Expression
ntrcls0	|- ((J e. Top /\ S C_ X /\ ((int` J)` ((cls` J)` S)) = (/)) -> ((int` J)` S) = (/))

Proof of Theorem ntrcls0

Step	Hyp	Ref	Expression
1		simpl 346	. . . . 5 |- ((J e. Top /\ S C_ X) -> J e. Top)
2		clscld.1	. . . . . 6 |- X = U.J
3	2	clsss3 8967	. . . . 5 |- ((J e. Top /\ S C_ X) -> ((cls` J)` S) C_ X)
4	2	sscls 8965	. . . . 5 |- ((J e. Top /\ S C_ X) -> S C_ ((cls` J)` S))
5	2	ntrss 8964	. . . . 5 |- ((J e. Top /\ ((cls` J)` S) C_ X /\ S C_ ((cls` J)` S)) -> ((int` J)` S) C_ ((int` J)` ((cls` J)` S)))
6	1, 3, 4, 5	syl111anc 1100	. . . 4 |- ((J e. Top /\ S C_ X) -> ((int` J)` S) C_ ((int` J)` ((cls` J)` S)))
7	6	3adant3 896	. . 3 |- ((J e. Top /\ S C_ X /\ ((int` J)` ((cls` J)` S)) = (/)) -> ((int` J)` S) C_ ((int` J)` ((cls` J)` S)))
8		sseq2 2639	. . . 4 |- (((int` J)` ((cls` J)` S)) = (/) -> (((int` J)` S) C_ ((int` J)` ((cls` J)` S)) <-> ((int` J)` S) C_ (/)))
9	8	3ad2ant3 899	. . 3 |- ((J e. Top /\ S C_ X /\ ((int` J)` ((cls` J)` S)) = (/)) -> (((int` J)` S) C_ ((int` J)` ((cls` J)` S)) <-> ((int` J)` S) C_ (/)))
10	7, 9	mpbid 212	. 2 |- ((J e. Top /\ S C_ X /\ ((int` J)` ((cls` J)` S)) = (/)) -> ((int` J)` S) C_ (/))
11		ss0 2902	. 2 |- (((int` J)` S) C_ (/) -> ((int` J)` S) = (/))
12	10, 11	syl 12	1 |- ((J e. Top /\ S C_ X /\ ((int` J)` ((cls` J)` S)) = (/)) -> ((int` J)` S) = (/))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   C_ wss 2593  (/)c0 2875  U.cuni 3177  ` cfv 3998  Topctop 8857  intcnt 8937  clsccl 8938

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-iin 3258  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-top 8861  df-cld 8939  df-ntr 8940  df-cls 8941

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