Theorem clsingemp 14961

Description: The closed sets of the topology {(/)}.

Assertion

Ref	Expression
clsingemp	|- (Clsd` {(/)}) = {(/)}

Proof of Theorem clsingemp

Step	Hyp	Ref	Expression
1		pw0 3132	. 2 |- ~P(/) = {(/)}
2		0ex 3446	. . . . . 6 |- (/) e. _V
3	2	dtopcl 14948	. . . . 5 |- ~P(/) = (Clsd` ~P(/))
4	3	eqcomi 1888	. . . 4 |- (Clsd` ~P(/)) = ~P(/)
5		fveq2 4681	. . . 4 |- ({(/)} = ~P(/) -> (Clsd` {(/)}) = (Clsd` ~P(/)))
6		id 73	. . . 4 |- ({(/)} = ~P(/) -> {(/)} = ~P(/))
7	4, 5, 6	3eqtr4a 1954	. . 3 |- ({(/)} = ~P(/) -> (Clsd` {(/)}) = {(/)})
8	7	eqcoms 1887	. 2 |- (~P(/) = {(/)} -> (Clsd` {(/)}) = {(/)})
9	1, 8	ax-mp 7	1 |- (Clsd` {(/)}) = {(/)}

Syntax hints:   = wceq 1298  (/)c0 2875  ~Pcpw 3032  {csn 3044  ` cfv 3998  Clsdccld 8936

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-rab 2112  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-top 8861  df-cld 8939

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