Theorem metn0 13172

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 13145	. . . . 5 |- ( D e. ( Met ` X ) -> D : ( X X. X ) --> RR )
2		frel 5352	. . . . 5 |- ( D : ( X X. X ) --> RR -> Rel D )
3		reldm0 4829	. . . . 5 |- ( Rel D -> ( D = (/) <-> dom D = (/) ) )
4	1, 2, 3	3syl 17	. . . 4 |- ( D e. ( Met ` X ) -> ( D = (/) <-> dom D = (/) ) )
5	1	fdmd 5354	. . . . 5 |- ( D e. ( Met ` X ) -> dom D = ( X X. X ) )
6	5	eqeq1d 2179	. . . 4 |- ( D e. ( Met ` X ) -> ( dom D = (/) <-> ( X X. X ) = (/) ) )
7	4, 6	bitrd 187	. . 3 |- ( D e. ( Met ` X ) -> ( D = (/) <-> ( X X. X ) = (/) ) )
8		sqxpeq0 5034	. . 3 |- ( ( X X. X ) = (/) <-> X = (/) )
9	7, 8	bitrdi 195	. 2 |- ( D e. ( Met ` X ) -> ( D = (/) <-> X = (/) ) )
10	9	necon3bid 2381	1 |- ( D e. ( Met ` X ) -> ( D =/= (/) <-> X =/= (/) ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   e. wcel 2141   =/= wne 2340   (/)c0 3414   X. cxp 4609   dom cdm 4611   Rel wrel 4616   -->wf 5194   ` cfv 5198   RRcr 7773   Metcmet 12775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-map 6628  df-met 12783

This theorem is referenced by: (None)