Theorem difopn 14695

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 3892	. . . . . 6 |- ( A e. J -> A C_ U. J )
2		iscld.1	. . . . . 6 |- X = U. J
3	1, 2	sseqtrrdi 3250	. . . . 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 3187	. . . 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 3298	. 2 |- ( ( A e. J /\ B e. ( Clsd ` J ) ) -> ( ( A i^i X ) \ B ) = ( A \ B ) )
8		indif2 3425	. . 3 |- ( A i^i ( X \ B ) ) = ( ( A i^i X ) \ B )
9		cldrcl 14689	. . . . 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 14694	. . . . 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 14590	. . . 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 2295	. 2 |- ( ( A e. J /\ B e. ( Clsd ` J ) ) -> ( ( A i^i X ) \ B ) e. J )
17	7, 16	eqeltrrd 2285	1 |- ( ( A e. J /\ B e. ( Clsd ` J ) ) -> ( A \ B ) e. J )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   \ cdif 3171   i^i cin 3173   C_ wss 3174   U.cuni 3864   ` cfv 5290   Topctop 14584   Clsdccld 14679

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298  df-top 14585  df-cld 14682

This theorem is referenced by: (None)