MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  metn0 Structured version   Visualization version   GIF version

Theorem metn0 24386
Description: A metric space is nonempty iff its base set is nonempty. (Contributed by NM, 4-Oct-2007.) (Revised by Mario Carneiro, 14-Aug-2015.)
Assertion
Ref Expression
metn0 (𝐷 ∈ (Met‘𝑋) → (𝐷 ≠ ∅ ↔ 𝑋 ≠ ∅))

Proof of Theorem metn0
StepHypRef Expression
1 metf 24356 . . . . 5 (𝐷 ∈ (Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ)
2 frel 6742 . . . . 5 (𝐷:(𝑋 × 𝑋)⟶ℝ → Rel 𝐷)
3 reldm0 5941 . . . . 5 (Rel 𝐷 → (𝐷 = ∅ ↔ dom 𝐷 = ∅))
41, 2, 33syl 18 . . . 4 (𝐷 ∈ (Met‘𝑋) → (𝐷 = ∅ ↔ dom 𝐷 = ∅))
51fdmd 6747 . . . . 5 (𝐷 ∈ (Met‘𝑋) → dom 𝐷 = (𝑋 × 𝑋))
65eqeq1d 2737 . . . 4 (𝐷 ∈ (Met‘𝑋) → (dom 𝐷 = ∅ ↔ (𝑋 × 𝑋) = ∅))
74, 6bitrd 279 . . 3 (𝐷 ∈ (Met‘𝑋) → (𝐷 = ∅ ↔ (𝑋 × 𝑋) = ∅))
8 xpeq0 6182 . . . 4 ((𝑋 × 𝑋) = ∅ ↔ (𝑋 = ∅ ∨ 𝑋 = ∅))
9 oridm 904 . . . 4 ((𝑋 = ∅ ∨ 𝑋 = ∅) ↔ 𝑋 = ∅)
108, 9bitri 275 . . 3 ((𝑋 × 𝑋) = ∅ ↔ 𝑋 = ∅)
117, 10bitrdi 287 . 2 (𝐷 ∈ (Met‘𝑋) → (𝐷 = ∅ ↔ 𝑋 = ∅))
1211necon3bid 2983 1 (𝐷 ∈ (Met‘𝑋) → (𝐷 ≠ ∅ ↔ 𝑋 ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wo 847   = wceq 1537  wcel 2106  wne 2938  c0 4339   × cxp 5687  dom cdm 5689  Rel wrel 5694  wf 6559  cfv 6563  cr 11152  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)
  Copyright terms: Public domain W3C validator