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Theorem 0met 22111
 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 4760 . 2 ∅ ∈ V
2 f0 6053 . . 3 ∅:∅⟶ℝ
3 xp0 5521 . . . 4 (∅ × ∅) = ∅
43feq2i 6004 . . 3 (∅:(∅ × ∅)⟶ℝ ↔ ∅:∅⟶ℝ)
52, 4mpbir 221 . 2 ∅:(∅ × ∅)⟶ℝ
6 noel 3901 . . . 4 ¬ 𝑥 ∈ ∅
76pm2.21i 116 . . 3 (𝑥 ∈ ∅ → ((𝑥𝑦) = 0 ↔ 𝑥 = 𝑦))
87adantr 481 . 2 ((𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅) → ((𝑥𝑦) = 0 ↔ 𝑥 = 𝑦))
96pm2.21i 116 . . 3 (𝑥 ∈ ∅ → (𝑥𝑦) ≤ ((𝑧𝑥) + (𝑧𝑦)))
1093ad2ant1 1080 . 2 ((𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) → (𝑥𝑦) ≤ ((𝑧𝑥) + (𝑧𝑦)))
111, 5, 8, 10ismeti 22070 1 ∅ ∈ (Met‘∅)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   = wceq 1480   ∈ wcel 1987  ∅c0 3897   class class class wbr 4623   × cxp 5082  ⟶wf 5853  ‘cfv 5857  (class class class)co 6615  ℝcr 9895  0cc0 9896   + caddc 9899   ≤ cle 10035  Metcme 19672 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-map 7819  df-met 19680 This theorem is referenced by: (None)
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