Theorem nrmsep2 21160

Description: In a normal space, any two disjoint closed sets have the property that each one is a subset of an open set whose closure is disjoint from the other. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 24-Aug-2015.)

Assertion

Ref	Expression
nrmsep2	|- ( ( J e. Nrm /\ ( C e. ( Clsd ` J ) /\ D e. ( Clsd ` J ) /\ ( C i^i D ) = (/) ) ) -> E. x e. J ( C C_ x /\ ( ( ( cls ` J ) ` x ) i^i D ) = (/) ) )

Distinct variable groups:   x, C   x, D   x, J

Proof of Theorem nrmsep2

Step	Hyp	Ref	Expression
1		simpl 473	. . 3 |- ( ( J e. Nrm /\ ( C e. ( Clsd ` J ) /\ D e. ( Clsd ` J ) /\ ( C i^i D ) = (/) ) ) -> J e. Nrm )
2		simpr2 1068	. . . 4 |- ( ( J e. Nrm /\ ( C e. ( Clsd ` J ) /\ D e. ( Clsd ` J ) /\ ( C i^i D ) = (/) ) ) -> D e. ( Clsd ` J ) )
3		eqid 2622	. . . . 5 |- U. J = U. J
4	3	cldopn 20835	. . . 4 |- ( D e. ( Clsd ` J ) -> ( U. J \ D ) e. J )
5	2, 4	syl 17	. . 3 |- ( ( J e. Nrm /\ ( C e. ( Clsd ` J ) /\ D e. ( Clsd ` J ) /\ ( C i^i D ) = (/) ) ) -> ( U. J \ D ) e. J )
6		simpr1 1067	. . 3 |- ( ( J e. Nrm /\ ( C e. ( Clsd ` J ) /\ D e. ( Clsd ` J ) /\ ( C i^i D ) = (/) ) ) -> C e. ( Clsd ` J ) )
7		simpr3 1069	. . . 4 |- ( ( J e. Nrm /\ ( C e. ( Clsd ` J ) /\ D e. ( Clsd ` J ) /\ ( C i^i D ) = (/) ) ) -> ( C i^i D ) = (/) )
8	3	cldss 20833	. . . . 5 |- ( C e. ( Clsd ` J ) -> C C_ U. J )
9		reldisj 4020	. . . . 5 |- ( C C_ U. J -> ( ( C i^i D ) = (/) <-> C C_ ( U. J \ D ) ) )
10	6, 8, 9	3syl 18	. . . 4 |- ( ( J e. Nrm /\ ( C e. ( Clsd ` J ) /\ D e. ( Clsd ` J ) /\ ( C i^i D ) = (/) ) ) -> ( ( C i^i D ) = (/) <-> C C_ ( U. J \ D ) ) )
11	7, 10	mpbid 222	. . 3 |- ( ( J e. Nrm /\ ( C e. ( Clsd ` J ) /\ D e. ( Clsd ` J ) /\ ( C i^i D ) = (/) ) ) -> C C_ ( U. J \ D ) )
12		nrmsep3 21159	. . 3 |- ( ( J e. Nrm /\ ( ( U. J \ D ) e. J /\ C e. ( Clsd ` J ) /\ C C_ ( U. J \ D ) ) ) -> E. x e. J ( C C_ x /\ ( ( cls ` J ) ` x ) C_ ( U. J \ D ) ) )
13	1, 5, 6, 11, 12	syl13anc 1328	. 2 |- ( ( J e. Nrm /\ ( C e. ( Clsd ` J ) /\ D e. ( Clsd ` J ) /\ ( C i^i D ) = (/) ) ) -> E. x e. J ( C C_ x /\ ( ( cls ` J ) ` x ) C_ ( U. J \ D ) ) )
14		ssdifin0 4050	. . . 4 |- ( ( ( cls ` J ) ` x ) C_ ( U. J \ D ) -> ( ( ( cls ` J ) ` x ) i^i D ) = (/) )
15	14	anim2i 593	. . 3 |- ( ( C C_ x /\ ( ( cls ` J ) ` x ) C_ ( U. J \ D ) ) -> ( C C_ x /\ ( ( ( cls ` J ) ` x ) i^i D ) = (/) ) )
16	15	reximi 3011	. 2 |- ( E. x e. J ( C C_ x /\ ( ( cls ` J ) ` x ) C_ ( U. J \ D ) ) -> E. x e. J ( C C_ x /\ ( ( ( cls ` J ) ` x ) i^i D ) = (/) ) )
17	13, 16	syl 17	1 |- ( ( J e. Nrm /\ ( C e. ( Clsd ` J ) /\ D e. ( Clsd ` J ) /\ ( C i^i D ) = (/) ) ) -> E. x e. J ( C C_ x /\ ( ( ( cls ` J ) ` x ) i^i D ) = (/) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   U.cuni 4436   ` cfv 5888   Clsdccld 20820   clsccl 20822   Nrmcnrm 21114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-top 20699  df-cld 20823  df-nrm 21121

This theorem is referenced by:  nrmsep  21161  isnrm2  21162