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Theorem 0met 22392
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 4995 . 2 ∅ ∈ V
2 f0 6308 . . 3 ∅:∅⟶ℝ
3 xp0 5774 . . . 4 (∅ × ∅) = ∅
43feq2i 6255 . . 3 (∅:(∅ × ∅)⟶ℝ ↔ ∅:∅⟶ℝ)
52, 4mpbir 222 . 2 ∅:(∅ × ∅)⟶ℝ
6 noel 4131 . . . 4 ¬ 𝑥 ∈ ∅
76pm2.21i 117 . . 3 (𝑥 ∈ ∅ → ((𝑥𝑦) = 0 ↔ 𝑥 = 𝑦))
87adantr 468 . 2 ((𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅) → ((𝑥𝑦) = 0 ↔ 𝑥 = 𝑦))
96pm2.21i 117 . . 3 (𝑥 ∈ ∅ → (𝑥𝑦) ≤ ((𝑧𝑥) + (𝑧𝑦)))
1093ad2ant1 1156 . 2 ((𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) → (𝑥𝑦) ≤ ((𝑧𝑥) + (𝑧𝑦)))
111, 5, 8, 10ismeti 22351 1 ∅ ∈ (Met‘∅)
Colors of variables: wff setvar class
Syntax hints:  wb 197   = wceq 1637  wcel 2157  c0 4127   class class class wbr 4855   × cxp 5320  wf 6104  cfv 6108  (class class class)co 6881  cr 10227  0cc0 10228   + caddc 10231  cle 10367  Metcme 19947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-sep 4986  ax-nul 4994  ax-pow 5046  ax-pr 5107  ax-un 7186  ax-cnex 10284  ax-resscn 10285
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3404  df-sbc 3645  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-br 4856  df-opab 4918  df-mpt 4935  df-id 5230  df-xp 5328  df-rel 5329  df-cnv 5330  df-co 5331  df-dm 5332  df-rn 5333  df-iota 6071  df-fun 6110  df-fn 6111  df-f 6112  df-fv 6116  df-ov 6884  df-oprab 6885  df-mpt2 6886  df-map 8101  df-met 19955
This theorem is referenced by: (None)
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