Theorem ntrcls0 14845

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 14844	. . . . 5 |- ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` S ) C_ X )
4	2	sscls 14834	. . . . 5 |- ( ( J e. Top /\ S C_ X ) -> S C_ ( ( cls ` J ) ` S ) )
5	2	ntrss 14833	. . . . 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 1271	. . . 4 |- ( ( J e. Top /\ S C_ X ) -> ( ( int ` J ) ` S ) C_ ( ( int ` J ) ` ( ( cls ` J ) ` S ) ) )
7	6	3adant3 1041	. . 3 |- ( ( J e. Top /\ S C_ X /\ ( ( int ` J ) ` ( ( cls ` J ) ` S ) ) = (/) ) -> ( ( int ` J ) ` S ) C_ ( ( int ` J ) ` ( ( cls ` J ) ` S ) ) )
8		sseq2 3249	. . . 4 |- ( ( ( int ` J ) ` ( ( cls ` J ) ` S ) ) = (/) -> ( ( ( int ` J ) ` S ) C_ ( ( int ` J ) ` ( ( cls ` J ) ` S ) ) <-> ( ( int ` J ) ` S ) C_ (/) ) )
9	8	3ad2ant3 1044	. . 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 3533	. 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 1002   = wceq 1395   e. wcel 2200   C_ wss 3198   (/)c0 3492   U.cuni 3891   ` cfv 5324   Topctop 14711   intcnt 14807   clsccl 14808

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-top 14712  df-cld 14809  df-ntr 14810  df-cls 14811

This theorem is referenced by: (None)