Theorem ntrcls0 14718

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 14717	. . . . 5 |- ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` S ) C_ X )
4	2	sscls 14707	. . . . 5 |- ( ( J e. Top /\ S C_ X ) -> S C_ ( ( cls ` J ) ` S ) )
5	2	ntrss 14706	. . . . 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 1250	. . . 4 |- ( ( J e. Top /\ S C_ X ) -> ( ( int ` J ) ` S ) C_ ( ( int ` J ) ` ( ( cls ` J ) ` S ) ) )
7	6	3adant3 1020	. . 3 |- ( ( J e. Top /\ S C_ X /\ ( ( int ` J ) ` ( ( cls ` J ) ` S ) ) = (/) ) -> ( ( int ` J ) ` S ) C_ ( ( int ` J ) ` ( ( cls ` J ) ` S ) ) )
8		sseq2 3225	. . . 4 |- ( ( ( int ` J ) ` ( ( cls ` J ) ` S ) ) = (/) -> ( ( ( int ` J ) ` S ) C_ ( ( int ` J ) ` ( ( cls ` J ) ` S ) ) <-> ( ( int ` J ) ` S ) C_ (/) ) )
9	8	3ad2ant3 1023	. . 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 3509	. 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 981   = wceq 1373   e. wcel 2178   C_ wss 3174   (/)c0 3468   U.cuni 3864   ` cfv 5290   Topctop 14584   intcnt 14680   clsccl 14681

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-top 14585  df-cld 14682  df-ntr 14683  df-cls 14684

This theorem is referenced by: (None)