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Theorem metn0 18351
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 18321 . . . . 5  |-  ( D  e.  ( Met `  X
)  ->  D :
( X  X.  X
) --> RR )
2 frel 5561 . . . . 5  |-  ( D : ( X  X.  X ) --> RR  ->  Rel 
D )
3 reldm0 5054 . . . . 5  |-  ( Rel 
D  ->  ( D  =  (/)  <->  dom  D  =  (/) ) )
41, 2, 33syl 19 . . . 4  |-  ( D  e.  ( Met `  X
)  ->  ( D  =  (/)  <->  dom  D  =  (/) ) )
5 fdm 5562 . . . . . 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 2420 . . . 4  |-  ( D  e.  ( Met `  X
)  ->  ( dom  D  =  (/)  <->  ( X  X.  X )  =  (/) ) )
84, 7bitrd 245 . . 3  |-  ( D  e.  ( Met `  X
)  ->  ( D  =  (/)  <->  ( X  X.  X )  =  (/) ) )
9 xpeq0 5260 . . . 4  |-  ( ( X  X.  X )  =  (/)  <->  ( X  =  (/)  \/  X  =  (/) ) )
10 oridm 501 . . . 4  |-  ( ( X  =  (/)  \/  X  =  (/) )  <->  X  =  (/) )
119, 10bitri 241 . . 3  |-  ( ( X  X.  X )  =  (/)  <->  X  =  (/) )
128, 11syl6bb 253 . 2  |-  ( D  e.  ( Met `  X
)  ->  ( D  =  (/)  <->  X  =  (/) ) )
1312necon3bid 2610 1  |-  ( D  e.  ( Met `  X
)  ->  ( D  =/=  (/)  <->  X  =/=  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    = wceq 1649    e. wcel 1721    =/= wne 2575   (/)c0 3596    X. cxp 4843   dom cdm 4845   Rel wrel 4850   -->wf 5417   ` cfv 5421   RRcr 8953   Metcme 16650
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-map 6987  df-met 16659
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