Theorem opnfss 9135

Description: The family of open sets of a metric space is a collection of subsets of the base set.

Hypotheses

Ref	Expression
opnfval.1	|- X = dom dom D
opnfval.2	|- J = (Open` D)

Assertion

Ref	Expression
opnfss	|- (D e. Met -> J C_ ~PX)

Proof of Theorem opnfss

Step	Hyp	Ref	Expression
1		opnfval.1	. . 3 |- X = dom dom D
2		opnfval.2	. . 3 |- J = (Open` D)
3	1, 2	opnfval 9134	. 2 |- (D e. Met -> J = {x | (x C_ X /\ A.y e. x E.z e. ran ( ball ` D)(y e. z /\ z C_ x))})
4		simpl 346	. . . . 5 |- ((x C_ X /\ A.y e. x E.z e. ran ( ball ` D)(y e. z /\ z C_ x)) -> x C_ X)
5	4	ss2abi 2679	. . . 4 |- {x | (x C_ X /\ A.y e. x E.z e. ran ( ball ` D)(y e. z /\ z C_ x))} C_ {x | x C_ X}
6		df-pw 3035	. . . 4 |- ~PX = {x | x C_ X}
7	5, 6	sseqtr4i 2650	. . 3 |- {x | (x C_ X /\ A.y e. x E.z e. ran ( ball ` D)(y e. z /\ z C_ x))} C_ ~PX
8	7	a1i 8	. 2 |- (D e. Met -> {x | (x C_ X /\ A.y e. x E.z e. ran ( ball ` D)(y e. z /\ z C_ x))} C_ ~PX)
9	3, 8	eqsstrd 2651	1 |- (D e. Met -> J C_ ~PX)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  A.wral 2105  E.wrex 2106   C_ wss 2593  ~Pcpw 3032  dom cdm 3986  ran crn 3987  ` cfv 3998  Metcme 9066   ball cbl 9068  Opencopn 9069

This theorem is referenced by:  uniopn2 9138  opnuni 9145  heiborlem18 15972

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-opn 9073

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