Theorem difopn 13693

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 3839	. . . . . 6 |- ( A e. J -> A C_ U. J )
2		iscld.1	. . . . . 6 |- X = U. J
3	1, 2	sseqtrrdi 3206	. . . . 5 |- ( A e. J -> A C_ X )
4	3	adantr 276	. . . 4 |- ( ( A e. J /\ B e. ( Clsd ` J ) ) -> A C_ X )
5		df-ss 3144	. . . 4 |- ( A C_ X <-> ( A i^i X ) = A )
6	4, 5	sylib 122	. . 3 |- ( ( A e. J /\ B e. ( Clsd ` J ) ) -> ( A i^i X ) = A )
7	6	difeq1d 3254	. 2 |- ( ( A e. J /\ B e. ( Clsd ` J ) ) -> ( ( A i^i X ) \ B ) = ( A \ B ) )
8		indif2 3381	. . 3 |- ( A i^i ( X \ B ) ) = ( ( A i^i X ) \ B )
9		cldrcl 13687	. . . . 5 |- ( B e. ( Clsd ` J ) -> J e. Top )
10	9	adantl 277	. . . 4 |- ( ( A e. J /\ B e. ( Clsd ` J ) ) -> J e. Top )
11		simpl 109	. . . 4 |- ( ( A e. J /\ B e. ( Clsd ` J ) ) -> A e. J )
12	2	cldopn 13692	. . . . 5 |- ( B e. ( Clsd ` J ) -> ( X \ B ) e. J )
13	12	adantl 277	. . . 4 |- ( ( A e. J /\ B e. ( Clsd ` J ) ) -> ( X \ B ) e. J )
14		inopn 13588	. . . 4 |- ( ( J e. Top /\ A e. J /\ ( X \ B ) e. J ) -> ( A i^i ( X \ B ) ) e. J )
15	10, 11, 13, 14	syl3anc 1238	. . 3 |- ( ( A e. J /\ B e. ( Clsd ` J ) ) -> ( A i^i ( X \ B ) ) e. J )
16	8, 15	eqeltrrid 2265	. 2 |- ( ( A e. J /\ B e. ( Clsd ` J ) ) -> ( ( A i^i X ) \ B ) e. J )
17	7, 16	eqeltrrd 2255	1 |- ( ( A e. J /\ B e. ( Clsd ` J ) ) -> ( A \ B ) e. J )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   \ cdif 3128   i^i cin 3130   C_ wss 3131   U.cuni 3811   ` cfv 5218   Topctop 13582   Clsdccld 13677

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-top 13583  df-cld 13680

This theorem is referenced by: (None)