Theorem metn0 9098

Description: A metric space is nonempty iff its base set is nonempty.

Hypothesis

Ref	Expression
metf.1	|- X = dom dom D

Assertion

Ref	Expression
metn0	|- (D e. Met -> (D =/= (/) <-> X =/= (/)))

Proof of Theorem metn0

Step	Hyp	Ref	Expression
1		metf.1	. . . . . 6 |- X = dom dom D
2	1	metf 9084	. . . . 5 |- (D e. Met -> D:(X X. X)-->RR)
3		frel 4566	. . . . 5 |- (D:(X X. X)-->RR -> Rel D)
4		reldm0 4176	. . . . 5 |- (Rel D -> (D = (/) <-> dom D = (/)))
5	2, 3, 4	3syl 24	. . . 4 |- (D e. Met -> (D = (/) <-> dom D = (/)))
6		fdm 4567	. . . . . 6 |- (D:(X X. X)-->RR -> dom D = (X X. X))
7		relxp 4088	. . . . . . 7 |- Rel (X X. X)
8		releq 4071	. . . . . . 7 |- (dom D = (X X. X) -> (Rel dom D <-> Rel (X X. X)))
9	7, 8	mpbiri 211	. . . . . 6 |- (dom D = (X X. X) -> Rel dom D)
10	6, 9	syl 12	. . . . 5 |- (D:(X X. X)-->RR -> Rel dom D)
11		reldm0 4176	. . . . 5 |- (Rel dom D -> (dom D = (/) <-> dom dom D = (/)))
12	2, 10, 11	3syl 24	. . . 4 |- (D e. Met -> (dom D = (/) <-> dom dom D = (/)))
13	5, 12	bitrd 587	. . 3 |- (D e. Met -> (D = (/) <-> dom dom D = (/)))
14	1	eqeq1i 1891	. . 3 |- (X = (/) <-> dom dom D = (/))
15	13, 14	syl6bbr 597	. 2 |- (D e. Met -> (D = (/) <-> X = (/)))
16	15	necon3bid 2035	1 |- (D e. Met -> (D =/= (/) <-> X =/= (/)))

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298   e. wcel 1300   =/= wne 2017  (/)c0 2875   X. cxp 3984  dom cdm 3986  Rel wrel 3991  -->wf 3994  RRcr 6385  Metcme 9066

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-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-met 9070

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