Theorem difopn 14580

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 3878	. . . . . 6 |- ( A e. J -> A C_ U. J )
2		iscld.1	. . . . . 6 |- X = U. J
3	1, 2	sseqtrrdi 3242	. . . . 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 3179	. . . 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 3290	. 2 |- ( ( A e. J /\ B e. ( Clsd ` J ) ) -> ( ( A i^i X ) \ B ) = ( A \ B ) )
8		indif2 3417	. . 3 |- ( A i^i ( X \ B ) ) = ( ( A i^i X ) \ B )
9		cldrcl 14574	. . . . 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 14579	. . . . 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 14475	. . . 4 |- ( ( J e. Top /\ A e. J /\ ( X \ B ) e. J ) -> ( A i^i ( X \ B ) ) e. J )
15	10, 11, 13, 14	syl3anc 1250	. . 3 |- ( ( A e. J /\ B e. ( Clsd ` J ) ) -> ( A i^i ( X \ B ) ) e. J )
16	8, 15	eqeltrrid 2293	. 2 |- ( ( A e. J /\ B e. ( Clsd ` J ) ) -> ( ( A i^i X ) \ B ) e. J )
17	7, 16	eqeltrrd 2283	1 |- ( ( A e. J /\ B e. ( Clsd ` J ) ) -> ( A \ B ) e. J )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   \ cdif 3163   i^i cin 3165   C_ wss 3166   U.cuni 3850   ` cfv 5271   Topctop 14469   Clsdccld 14564

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-iota 5232  df-fun 5273  df-fn 5274  df-fv 5279  df-top 14470  df-cld 14567

This theorem is referenced by: (None)