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Theorem metn0 20051
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 20021 . . . . 5  |-  ( D  e.  ( Met `  X
)  ->  D :
( X  X.  X
) --> RR )
2 frel 5660 . . . . 5  |-  ( D : ( X  X.  X ) --> RR  ->  Rel 
D )
3 reldm0 5155 . . . . 5  |-  ( Rel 
D  ->  ( D  =  (/)  <->  dom  D  =  (/) ) )
41, 2, 33syl 20 . . . 4  |-  ( D  e.  ( Met `  X
)  ->  ( D  =  (/)  <->  dom  D  =  (/) ) )
5 fdm 5661 . . . . . 6  |-  ( D : ( X  X.  X ) --> RR  ->  dom 
D  =  ( X  X.  X ) )
61, 5syl 16 . . . . 5  |-  ( D  e.  ( Met `  X
)  ->  dom  D  =  ( X  X.  X
) )
76eqeq1d 2453 . . . 4  |-  ( D  e.  ( Met `  X
)  ->  ( dom  D  =  (/)  <->  ( X  X.  X )  =  (/) ) )
84, 7bitrd 253 . . 3  |-  ( D  e.  ( Met `  X
)  ->  ( D  =  (/)  <->  ( X  X.  X )  =  (/) ) )
9 xpeq0 5356 . . . 4  |-  ( ( X  X.  X )  =  (/)  <->  ( X  =  (/)  \/  X  =  (/) ) )
10 oridm 514 . . . 4  |-  ( ( X  =  (/)  \/  X  =  (/) )  <->  X  =  (/) )
119, 10bitri 249 . . 3  |-  ( ( X  X.  X )  =  (/)  <->  X  =  (/) )
128, 11syl6bb 261 . 2  |-  ( D  e.  ( Met `  X
)  ->  ( D  =  (/)  <->  X  =  (/) ) )
1312necon3bid 2706 1  |-  ( D  e.  ( Met `  X
)  ->  ( D  =/=  (/)  <->  X  =/=  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    = wceq 1370    e. wcel 1758    =/= wne 2644   (/)c0 3735    X. cxp 4936   dom cdm 4938   Rel wrel 4943   -->wf 5512   ` cfv 5516   RRcr 9382   Metcme 17911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-fv 5524  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-map 7316  df-met 17920
This theorem is referenced by: (None)
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