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Theorem 0met 24376
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 5307 . 2 ∅ ∈ V
2 f0 6789 . . 3 ∅:∅⟶ℝ
3 xp0 6178 . . . 4 (∅ × ∅) = ∅
43feq2i 6728 . . 3 (∅:(∅ × ∅)⟶ℝ ↔ ∅:∅⟶ℝ)
52, 4mpbir 231 . 2 ∅:(∅ × ∅)⟶ℝ
6 noel 4338 . . . 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 1134 . 2 ((𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) → (𝑥𝑦) ≤ ((𝑧𝑥) + (𝑧𝑦)))
111, 5, 8, 10ismeti 24335 1 ∅ ∈ (Met‘∅)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1540  wcel 2108  c0 4333   class class class wbr 5143   × cxp 5683  wf 6557  cfv 6561  (class class class)co 7431  cr 11154  0cc0 11155   + caddc 11158  cle 11296  Metcmet 21350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-met 21358
This theorem is referenced by: (None)
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