Theorem metn0 18395

Description: A metric space is nonempty iff its base set is nonempty. (Contributed by NM, 4-Oct-2007.) (Revised by Mario Carneiro, 14-Aug-2015.)

Assertion

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

Proof of Theorem metn0

Step	Hyp	Ref	Expression
1		metf 18365	. . . . 5 |- ( D e. ( Met ` X ) -> D : ( X X. X ) --> RR )
2		frel 5597	. . . . 5 |- ( D : ( X X. X ) --> RR -> Rel D )
3		reldm0 5090	. . . . 5 |- ( Rel D -> ( D = (/) <-> dom D = (/) ) )
4	1, 2, 3	3syl 19	. . . 4 |- ( D e. ( Met ` X ) -> ( D = (/) <-> dom D = (/) ) )
5		fdm 5598	. . . . . 6 |- ( D : ( X X. X ) --> RR -> dom D = ( X X. X ) )
6	1, 5	syl 16	. . . . 5 |- ( D e. ( Met ` X ) -> dom D = ( X X. X ) )
7	6	eqeq1d 2446	. . . 4 |- ( D e. ( Met ` X ) -> ( dom D = (/) <-> ( X X. X ) = (/) ) )
8	4, 7	bitrd 246	. . 3 |- ( D e. ( Met ` X ) -> ( D = (/) <-> ( X X. X ) = (/) ) )
9		xpeq0 5296	. . . 4 |- ( ( X X. X ) = (/) <-> ( X = (/) \/ X = (/) ) )
10		oridm 502	. . . 4 |- ( ( X = (/) \/ X = (/) ) <-> X = (/) )
11	9, 10	bitri 242	. . 3 |- ( ( X X. X ) = (/) <-> X = (/) )
12	8, 11	syl6bb 254	. 2 |- ( D e. ( Met ` X ) -> ( D = (/) <-> X = (/) ) )
13	12	necon3bid 2638	1 |- ( D e. ( Met ` X ) -> ( D =/= (/) <-> X =/= (/) ) )

Syntax hints:   -> wi 4   <-> wb 178   \/ wo 359   = wceq 1653   e. wcel 1726   =/= wne 2601   (/)c0 3630   X. cxp 4879   dom cdm 4881   Rel wrel 4886   -->wf 5453   ` cfv 5457   RRcr 8994   Metcme 16692

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-map 7023  df-met 16701