Theorem difopn 12902

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 3824	. . . . . 6 |- ( A e. J -> A C_ U. J )
2		iscld.1	. . . . . 6 |- X = U. J
3	1, 2	sseqtrrdi 3196	. . . . 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 3134	. . . 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 3244	. 2 |- ( ( A e. J /\ B e. ( Clsd ` J ) ) -> ( ( A i^i X ) \ B ) = ( A \ B ) )
8		indif2 3371	. . 3 |- ( A i^i ( X \ B ) ) = ( ( A i^i X ) \ B )
9		cldrcl 12896	. . . . 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 12901	. . . . 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 12795	. . . 4 |- ( ( J e. Top /\ A e. J /\ ( X \ B ) e. J ) -> ( A i^i ( X \ B ) ) e. J )
15	10, 11, 13, 14	syl3anc 1233	. . 3 |- ( ( A e. J /\ B e. ( Clsd ` J ) ) -> ( A i^i ( X \ B ) ) e. J )
16	8, 15	eqeltrrid 2258	. 2 |- ( ( A e. J /\ B e. ( Clsd ` J ) ) -> ( ( A i^i X ) \ B ) e. J )
17	7, 16	eqeltrrd 2248	1 |- ( ( A e. J /\ B e. ( Clsd ` J ) ) -> ( A \ B ) e. J )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   \ cdif 3118   i^i cin 3120   C_ wss 3121   U.cuni 3796   ` cfv 5198   Topctop 12789   Clsdccld 12886

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206  df-top 12790  df-cld 12889

This theorem is referenced by: (None)