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Theorem xmet0 22475
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 ∈ (∞Met‘𝑋) ∧ 𝐴𝑋) → (𝐴𝐷𝐴) = 0)

Proof of Theorem xmet0
StepHypRef Expression
1 eqid 2799 . 2 𝐴 = 𝐴
2 xmeteq0 22471 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴𝑋𝐴𝑋) → ((𝐴𝐷𝐴) = 0 ↔ 𝐴 = 𝐴))
323anidm23 1545 . 2 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴𝑋) → ((𝐴𝐷𝐴) = 0 ↔ 𝐴 = 𝐴))
41, 3mpbiri 250 1 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴𝑋) → (𝐴𝐷𝐴) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  cfv 6101  (class class class)co 6878  0cc0 10224  ∞Metcxmet 20053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-map 8097  df-xr 10367  df-xmet 20061
This theorem is referenced by:  met0  22476  xmetge0  22477  xmetsym  22480  xmetpsmet  22481  xblcntr  22544  ssbl  22556  xmeter  22566  ubthlem2  28252  sitmcl  30929
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