Theorem difopn 12277

Description: The difference of a closed set with an open set is open. (Contributed by Mario Carneiro, 6-Jan-2014.)

Hypothesis

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

Assertion

Ref	Expression
difopn	|- ( ( A e. J /\ B e. ( Clsd ` J ) ) -> ( A \ B ) e. J )

Proof of Theorem difopn

Step	Hyp	Ref	Expression
1		elssuni 3764	. . . . . 6 |- ( A e. J -> A C_ U. J )
2		iscld.1	. . . . . 6 |- X = U. J
3	1, 2	sseqtrrdi 3146	. . . . 5 |- ( A e. J -> A C_ X )
4	3	adantr 274	. . . 4 |- ( ( A e. J /\ B e. ( Clsd ` J ) ) -> A C_ X )
5		df-ss 3084	. . . 4 |- ( A C_ X <-> ( A i^i X ) = A )
6	4, 5	sylib 121	. . 3 |- ( ( A e. J /\ B e. ( Clsd ` J ) ) -> ( A i^i X ) = A )
7	6	difeq1d 3193	. 2 |- ( ( A e. J /\ B e. ( Clsd ` J ) ) -> ( ( A i^i X ) \ B ) = ( A \ B ) )
8		indif2 3320	. . 3 |- ( A i^i ( X \ B ) ) = ( ( A i^i X ) \ B )
9		cldrcl 12271	. . . . 5 |- ( B e. ( Clsd ` J ) -> J e. Top )
10	9	adantl 275	. . . 4 |- ( ( A e. J /\ B e. ( Clsd ` J ) ) -> J e. Top )
11		simpl 108	. . . 4 |- ( ( A e. J /\ B e. ( Clsd ` J ) ) -> A e. J )
12	2	cldopn 12276	. . . . 5 |- ( B e. ( Clsd ` J ) -> ( X \ B ) e. J )
13	12	adantl 275	. . . 4 |- ( ( A e. J /\ B e. ( Clsd ` J ) ) -> ( X \ B ) e. J )
14		inopn 12170	. . . 4 |- ( ( J e. Top /\ A e. J /\ ( X \ B ) e. J ) -> ( A i^i ( X \ B ) ) e. J )
15	10, 11, 13, 14	syl3anc 1216	. . 3 |- ( ( A e. J /\ B e. ( Clsd ` J ) ) -> ( A i^i ( X \ B ) ) e. J )
16	8, 15	eqeltrrid 2227	. 2 |- ( ( A e. J /\ B e. ( Clsd ` J ) ) -> ( ( A i^i X ) \ B ) e. J )
17	7, 16	eqeltrrd 2217	1 |- ( ( A e. J /\ B e. ( Clsd ` J ) ) -> ( A \ B ) e. J )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   \ cdif 3068   i^i cin 3070   C_ wss 3071   U.cuni 3736   ` cfv 5123   Topctop 12164   Clsdccld 12261

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fn 5126  df-fv 5131  df-top 12165  df-cld 12264

This theorem is referenced by: (None)