Theorem metn0 17924

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 17895	. . . . 5 |- ( D e. ( Met ` X ) -> D : ( X X. X ) --> RR )
2		frel 5392	. . . . 5 |- ( D : ( X X. X ) --> RR -> Rel D )
3		reldm0 4896	. . . . 5 |- ( Rel D -> ( D = (/) <-> dom D = (/) ) )
4	1, 2, 3	3syl 18	. . . 4 |- ( D e. ( Met ` X ) -> ( D = (/) <-> dom D = (/) ) )
5		fdm 5393	. . . . . 6 |- ( D : ( X X. X ) --> RR -> dom D = ( X X. X ) )
6	1, 5	syl 15	. . . . 5 |- ( D e. ( Met ` X ) -> dom D = ( X X. X ) )
7	6	eqeq1d 2291	. . . 4 |- ( D e. ( Met ` X ) -> ( dom D = (/) <-> ( X X. X ) = (/) ) )
8	4, 7	bitrd 244	. . 3 |- ( D e. ( Met ` X ) -> ( D = (/) <-> ( X X. X ) = (/) ) )
9		xpeq0 5100	. . . 4 |- ( ( X X. X ) = (/) <-> ( X = (/) \/ X = (/) ) )
10		oridm 500	. . . 4 |- ( ( X = (/) \/ X = (/) ) <-> X = (/) )
11	9, 10	bitri 240	. . 3 |- ( ( X X. X ) = (/) <-> X = (/) )
12	8, 11	syl6bb 252	. 2 |- ( D e. ( Met ` X ) -> ( D = (/) <-> X = (/) ) )
13	12	necon3bid 2481	1 |- ( D e. ( Met ` X ) -> ( D =/= (/) <-> X =/= (/) ) )

Syntax hints:   -> wi 4   <-> wb 176   \/ wo 357   = wceq 1623   e. wcel 1684   =/= wne 2446   (/)c0 3455   X. cxp 4687   dom cdm 4689   Rel wrel 4694   -->wf 5251   ` cfv 5255   RRcr 8736   Metcme 16370

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-met 16374