Theorem ssntr 15149

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 3412	. . . . 5 |- ( O e. ( J i^i ~P S ) <-> ( O e. J /\ O e. ~P S ) )
2		elpwg 3696	. . . . . 6 |- ( O e. J -> ( O e. ~P S <-> O C_ S ) )
3	2	pm5.32i 458	. . . . 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 3961	. . . 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 15137	. . 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 3287	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 1402   e. wcel 2209   i^i cin 3219   C_ wss 3220   ~Pcpw 3688   U.cuni 3933   ` cfv 5375   Topctop 15024   intcnt 15120

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 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 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  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-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-top 15025  df-ntr 15123

This theorem is referenced by:  ntrin  15151  neiint  15172  cnntri  15251