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Theorem xmet0 22870
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 2826 . 2 𝐴 = 𝐴
2 xmeteq0 22866 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴𝑋𝐴𝑋) → ((𝐴𝐷𝐴) = 0 ↔ 𝐴 = 𝐴))
323anidm23 1415 . 2 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴𝑋) → ((𝐴𝐷𝐴) = 0 ↔ 𝐴 = 𝐴))
41, 3mpbiri 259 1 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴𝑋) → (𝐴𝐷𝐴) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  cfv 6352  (class class class)co 7148  0cc0 10526  ∞Metcxmet 20449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-cnex 10582  ax-resscn 10583
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-fv 6360  df-ov 7151  df-oprab 7152  df-mpo 7153  df-map 8398  df-xr 10668  df-xmet 20457
This theorem is referenced by:  met0  22871  xmetge0  22872  xmetsym  22875  xmetpsmet  22876  xblcntr  22939  ssbl  22951  xmeter  22961  ubthlem2  28565  sitmcl  31498
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