Theorem metne0 7821

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

Hypothesis

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

Assertion

Ref	Expression
metne0	|- (D e. Met -> (D =/= (/) <-> X =/= (/)))

Proof of Theorem metne0

Step	Hyp	Ref	Expression
1		metf.1	. . . . . 6 |- X = dom dom D
2	1	metf 7807	. . . . 5 |- (D e. Met -> D:(X X. X)-->RR)
3		frel 3630	. . . . 5 |- (D:(X X. X)-->RR -> Rel D)
4		reldm0 3331	. . . . 5 |- (Rel D -> (D = (/) <-> dom D = (/)))
5	2, 3, 4	3syl 20	. . . 4 |- (D e. Met -> (D = (/) <-> dom D = (/)))
6		fdm 3631	. . . . . 6 |- (D:(X X. X)-->RR -> dom D = (X X. X))
7		relxp 3255	. . . . . . 7 |- Rel (X X. X)
8		releq 3243	. . . . . . 7 |- (dom D = (X X. X) -> (Rel dom D <-> Rel (X X. X)))
9	7, 8	mpbiri 194	. . . . . 6 |- (dom D = (X X. X) -> Rel dom D)
10	6, 9	syl 10	. . . . 5 |- (D:(X X. X)-->RR -> Rel dom D)
11		reldm0 3331	. . . . 5 |- (Rel dom D -> (dom D = (/) <-> dom dom D = (/)))
12	2, 10, 11	3syl 20	. . . 4 |- (D e. Met -> (dom D = (/) <-> dom dom D = (/)))
13	5, 12	bitrd 528	. . 3 |- (D e. Met -> (D = (/) <-> dom dom D = (/)))
14	1	eqeq1i 1482	. . 3 |- (X = (/) <-> dom dom D = (/))
15	13, 14	syl6bbr 538	. 2 |- (D e. Met -> (D = (/) <-> X = (/)))
16	15	necon3bid 1601	1 |- (D e. Met -> (D =/= (/) <-> X =/= (/)))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956   e. wcel 958   =/= wne 1585  (/)c0 2280   X. cxp 3168  dom cdm 3170  Rel wrel 3175  -->wf 3178  RRcr 5233  Metcme 7789

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198  df-opr 3965  df-met 7793

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