MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  metn0 Unicode version

Theorem metn0 18137
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 18108 . . . . 5  |-  ( D  e.  ( Met `  X
)  ->  D :
( X  X.  X
) --> RR )
2 frel 5498 . . . . 5  |-  ( D : ( X  X.  X ) --> RR  ->  Rel 
D )
3 reldm0 4999 . . . . 5  |-  ( Rel 
D  ->  ( D  =  (/)  <->  dom  D  =  (/) ) )
41, 2, 33syl 18 . . . 4  |-  ( D  e.  ( Met `  X
)  ->  ( D  =  (/)  <->  dom  D  =  (/) ) )
5 fdm 5499 . . . . . 6  |-  ( D : ( X  X.  X ) --> RR  ->  dom 
D  =  ( X  X.  X ) )
61, 5syl 15 . . . . 5  |-  ( D  e.  ( Met `  X
)  ->  dom  D  =  ( X  X.  X
) )
76eqeq1d 2374 . . . 4  |-  ( D  e.  ( Met `  X
)  ->  ( dom  D  =  (/)  <->  ( X  X.  X )  =  (/) ) )
84, 7bitrd 244 . . 3  |-  ( D  e.  ( Met `  X
)  ->  ( D  =  (/)  <->  ( X  X.  X )  =  (/) ) )
9 xpeq0 5203 . . . 4  |-  ( ( X  X.  X )  =  (/)  <->  ( X  =  (/)  \/  X  =  (/) ) )
10 oridm 500 . . . 4  |-  ( ( X  =  (/)  \/  X  =  (/) )  <->  X  =  (/) )
119, 10bitri 240 . . 3  |-  ( ( X  X.  X )  =  (/)  <->  X  =  (/) )
128, 11syl6bb 252 . 2  |-  ( D  e.  ( Met `  X
)  ->  ( D  =  (/)  <->  X  =  (/) ) )
1312necon3bid 2564 1  |-  ( D  e.  ( Met `  X
)  ->  ( D  =/=  (/)  <->  X  =/=  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    = wceq 1647    e. wcel 1715    =/= wne 2529   (/)c0 3543    X. cxp 4790   dom cdm 4792   Rel wrel 4797   -->wf 5354   ` cfv 5358   RRcr 8883   Metcme 16580
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-map 6917  df-met 16587
  Copyright terms: Public domain W3C validator