Theorem metn0 15479

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 15452	. . . . 5 |- ( D e. ( Met ` X ) -> D : ( X X. X ) --> RR )
2		frel 5538	. . . . 5 |- ( D : ( X X. X ) --> RR -> Rel D )
3		reldm0 4999	. . . . 5 |- ( Rel D -> ( D = (/) <-> dom D = (/) ) )
4	1, 2, 3	3syl 17	. . . 4 |- ( D e. ( Met ` X ) -> ( D = (/) <-> dom D = (/) ) )
5	1	fdmd 5540	. . . . 5 |- ( D e. ( Met ` X ) -> dom D = ( X X. X ) )
6	5	eqeq1d 2247	. . . 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 5211	. . 3 |- ( ( X X. X ) = (/) <-> X = (/) )
9	7, 8	bitrdi 196	. 2 |- ( D e. ( Met ` X ) -> ( D = (/) <-> X = (/) ) )
10	9	necon3bid 2461	1 |- ( D e. ( Met ` X ) -> ( D =/= (/) <-> X =/= (/) ) )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   = wceq 1402   e. wcel 2209   =/= wne 2420   (/)c0 3520   X. cxp 4772   dom cdm 4774   Rel wrel 4779   -->wf 5373   ` cfv 5377   RRcr 8178   Metcmet 14874

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-map 6924  df-met 14882

This theorem is used by: (None)