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Theorem metn0 12778
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
StepHypRef Expression
1 metf 12751 . . . . 5  |-  ( D  e.  ( Met `  X
)  ->  D :
( X  X.  X
) --> RR )
2 frel 5324 . . . . 5  |-  ( D : ( X  X.  X ) --> RR  ->  Rel 
D )
3 reldm0 4804 . . . . 5  |-  ( Rel 
D  ->  ( D  =  (/)  <->  dom  D  =  (/) ) )
41, 2, 33syl 17 . . . 4  |-  ( D  e.  ( Met `  X
)  ->  ( D  =  (/)  <->  dom  D  =  (/) ) )
51fdmd 5326 . . . . 5  |-  ( D  e.  ( Met `  X
)  ->  dom  D  =  ( X  X.  X
) )
65eqeq1d 2166 . . . 4  |-  ( D  e.  ( Met `  X
)  ->  ( dom  D  =  (/)  <->  ( X  X.  X )  =  (/) ) )
74, 6bitrd 187 . . 3  |-  ( D  e.  ( Met `  X
)  ->  ( D  =  (/)  <->  ( X  X.  X )  =  (/) ) )
8 sqxpeq0 5009 . . 3  |-  ( ( X  X.  X )  =  (/)  <->  X  =  (/) )
97, 8bitrdi 195 . 2  |-  ( D  e.  ( Met `  X
)  ->  ( D  =  (/)  <->  X  =  (/) ) )
109necon3bid 2368 1  |-  ( D  e.  ( Met `  X
)  ->  ( D  =/=  (/)  <->  X  =/=  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1335    e. wcel 2128    =/= wne 2327   (/)c0 3394    X. cxp 4584   dom cdm 4586   Rel wrel 4591   -->wf 5166   ` cfv 5170   RRcr 7731   Metcmet 12381
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-fv 5178  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-map 6595  df-met 12389
This theorem is referenced by: (None)
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