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Theorem 0met 24392
Description: The empty metric. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
Assertion
Ref Expression
0met ∅ ∈ (Met‘∅)

Proof of Theorem 0met
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ex 5313 . 2 ∅ ∈ V
2 f0 6790 . . 3 ∅:∅⟶ℝ
3 xp0 6180 . . . 4 (∅ × ∅) = ∅
43feq2i 6729 . . 3 (∅:(∅ × ∅)⟶ℝ ↔ ∅:∅⟶ℝ)
52, 4mpbir 231 . 2 ∅:(∅ × ∅)⟶ℝ
6 noel 4344 . . . 4 ¬ 𝑥 ∈ ∅
76pm2.21i 119 . . 3 (𝑥 ∈ ∅ → ((𝑥𝑦) = 0 ↔ 𝑥 = 𝑦))
87adantr 480 . 2 ((𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅) → ((𝑥𝑦) = 0 ↔ 𝑥 = 𝑦))
96pm2.21i 119 . . 3 (𝑥 ∈ ∅ → (𝑥𝑦) ≤ ((𝑧𝑥) + (𝑧𝑦)))
1093ad2ant1 1132 . 2 ((𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) → (𝑥𝑦) ≤ ((𝑧𝑥) + (𝑧𝑦)))
111, 5, 8, 10ismeti 24351 1 ∅ ∈ (Met‘∅)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1537  wcel 2106  c0 4339   class class class wbr 5148   × cxp 5687  wf 6559  cfv 6563  (class class class)co 7431  cr 11152  0cc0 11153   + caddc 11156  cle 11294  Metcmet 21368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-met 21376
This theorem is referenced by: (None)
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