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Theorem xmet0 20608
Description: The distance function of a metric space is zero if its arguments are equal. Definition 14-1.1(a) of [Gleason] p. 223. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xmet0  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X
)  ->  ( A D A )  =  0 )

Proof of Theorem xmet0
StepHypRef Expression
1 eqid 2467 . 2  |-  A  =  A
2 xmeteq0 20604 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  A  e.  X
)  ->  ( ( A D A )  =  0  <->  A  =  A
) )
323anidm23 1287 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X
)  ->  ( ( A D A )  =  0  <->  A  =  A
) )
41, 3mpbiri 233 1  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X
)  ->  ( A D A )  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   ` cfv 5588  (class class class)co 6284   0cc0 9492   *Metcxmt 18202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-map 7422  df-xr 9632  df-xmet 18211
This theorem is referenced by:  met0  20609  xmetge0  20610  xmetsym  20613  xmetpsmet  20614  xblcntr  20677  ssbl  20689  xmeter  20699  metustidOLD  20825  ubthlem2  25491
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