Theorem ssntr 14796

Description: An open subset of a set is a subset of the set's interior. (Contributed by Jeff Hankins, 31-Aug-2009.) (Revised by Mario Carneiro, 11-Nov-2013.)

Hypothesis

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

Assertion

Ref	Expression
ssntr	|- ( ( ( J e. Top /\ S C_ X ) /\ ( O e. J /\ O C_ S ) ) -> O C_ ( ( int ` J ) ` S ) )

Proof of Theorem ssntr

Step	Hyp	Ref	Expression
1		elin 3387	. . . . 5 |- ( O e. ( J i^i ~P S ) <-> ( O e. J /\ O e. ~P S ) )
2		elpwg 3657	. . . . . 6 |- ( O e. J -> ( O e. ~P S <-> O C_ S ) )
3	2	pm5.32i 454	. . . . 5 |- ( ( O e. J /\ O e. ~P S ) <-> ( O e. J /\ O C_ S ) )
4	1, 3	bitr2i 185	. . . 4 |- ( ( O e. J /\ O C_ S ) <-> O e. ( J i^i ~P S ) )
5		elssuni 3916	. . . 4 |- ( O e. ( J i^i ~P S ) -> O C_ U. ( J i^i ~P S ) )
6	4, 5	sylbi 121	. . 3 |- ( ( O e. J /\ O C_ S ) -> O C_ U. ( J i^i ~P S ) )
7	6	adantl 277	. 2 |- ( ( ( J e. Top /\ S C_ X ) /\ ( O e. J /\ O C_ S ) ) -> O C_ U. ( J i^i ~P S ) )
8		clscld.1	. . . 4 |- X = U. J
9	8	ntrval 14784	. . 3 |- ( ( J e. Top /\ S C_ X ) -> ( ( int ` J ) ` S ) = U. ( J i^i ~P S ) )
10	9	adantr 276	. 2 |- ( ( ( J e. Top /\ S C_ X ) /\ ( O e. J /\ O C_ S ) ) -> ( ( int ` J ) ` S ) = U. ( J i^i ~P S ) )
11	7, 10	sseqtrrd 3263	1 |- ( ( ( J e. Top /\ S C_ X ) /\ ( O e. J /\ O C_ S ) ) -> O C_ ( ( int ` J ) ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   i^i cin 3196   C_ wss 3197   ~Pcpw 3649   U.cuni 3888   ` cfv 5318   Topctop 14671   intcnt 14767

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-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 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  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 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-top 14672  df-ntr 14770

This theorem is referenced by:  ntrin  14798  neiint  14819  cnntri  14898