Theorem ntrcls0 15155

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 = 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 109	. . . . 5 |- ( ( J e. Top /\ S C_ X ) -> J e. Top )
2		clscld.1	. . . . . 6 |- X = U. J
3	2	clsss3 15154	. . . . 5 |- ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` S ) C_ X )
4	2	sscls 15144	. . . . 5 |- ( ( J e. Top /\ S C_ X ) -> S C_ ( ( cls ` J ) ` S ) )
5	2	ntrss 15143	. . . . 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	syl3anc 1278	. . . 4 |- ( ( J e. Top /\ S C_ X ) -> ( ( int ` J ) ` S ) C_ ( ( int ` J ) ` ( ( cls ` J ) ` S ) ) )
7	6	3adant3 1048	. . 3 |- ( ( J e. Top /\ S C_ X /\ ( ( int ` J ) ` ( ( cls ` J ) ` S ) ) = (/) ) -> ( ( int ` J ) ` S ) C_ ( ( int ` J ) ` ( ( cls ` J ) ` S ) ) )
8		sseq2 3272	. . . 4 |- ( ( ( int ` J ) ` ( ( cls ` J ) ` S ) ) = (/) -> ( ( ( int ` J ) ` S ) C_ ( ( int ` J ) ` ( ( cls ` J ) ` S ) ) <-> ( ( int ` J ) ` S ) C_ (/) ) )
9	8	3ad2ant3 1051	. . 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 147	. 2 |- ( ( J e. Top /\ S C_ X /\ ( ( int ` J ) ` ( ( cls ` J ) ` S ) ) = (/) ) -> ( ( int ` J ) ` S ) C_ (/) )
11		ss0 3563	. 2 |- ( ( ( int ` J ) ` S ) C_ (/) -> ( ( int ` J ) ` S ) = (/) )
12	10, 11	syl 14	1 |- ( ( J e. Top /\ S C_ X /\ ( ( int ` J ) ` ( ( cls ` J ) ` S ) ) = (/) ) -> ( ( int ` J ) ` S ) = (/) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   C_ wss 3220   (/)c0 3520   U.cuni 3930   ` cfv 5372   Topctop 15021   intcnt 15117   clsccl 15118

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-top 15022  df-cld 15119  df-ntr 15120  df-cls 15121

This theorem is referenced by: (None)