Theorem metn0 15046

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 15019	. . . . 5 |- ( D e. ( Met ` X ) -> D : ( X X. X ) --> RR )
2		frel 5477	. . . . 5 |- ( D : ( X X. X ) --> RR -> Rel D )
3		reldm0 4940	. . . . 5 |- ( Rel D -> ( D = (/) <-> dom D = (/) ) )
4	1, 2, 3	3syl 17	. . . 4 |- ( D e. ( Met ` X ) -> ( D = (/) <-> dom D = (/) ) )
5	1	fdmd 5479	. . . . 5 |- ( D e. ( Met ` X ) -> dom D = ( X X. X ) )
6	5	eqeq1d 2238	. . . 4 |- ( D e. ( Met ` X ) -> ( dom D = (/) <-> ( X X. X ) = (/) ) )
7	4, 6	bitrd 188	. . 3 |- ( D e. ( Met ` X ) -> ( D = (/) <-> ( X X. X ) = (/) ) )
8		sqxpeq0 5151	. . 3 |- ( ( X X. X ) = (/) <-> X = (/) )
9	7, 8	bitrdi 196	. 2 |- ( D e. ( Met ` X ) -> ( D = (/) <-> X = (/) ) )
10	9	necon3bid 2441	1 |- ( D e. ( Met ` X ) -> ( D =/= (/) <-> X =/= (/) ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   e. wcel 2200   =/= wne 2400   (/)c0 3491   X. cxp 4716   dom cdm 4718   Rel wrel 4723   -->wf 5313   ` cfv 5317   RRcr 7994   Metcmet 14495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-map 6795  df-met 14503

This theorem is referenced by: (None)