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Theorem metn0 22212
 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 22182 . . . . 5 (𝐷 ∈ (Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ)
2 frel 6088 . . . . 5 (𝐷:(𝑋 × 𝑋)⟶ℝ → Rel 𝐷)
3 reldm0 5375 . . . . 5 (Rel 𝐷 → (𝐷 = ∅ ↔ dom 𝐷 = ∅))
41, 2, 33syl 18 . . . 4 (𝐷 ∈ (Met‘𝑋) → (𝐷 = ∅ ↔ dom 𝐷 = ∅))
5 fdm 6089 . . . . . 6 (𝐷:(𝑋 × 𝑋)⟶ℝ → dom 𝐷 = (𝑋 × 𝑋))
61, 5syl 17 . . . . 5 (𝐷 ∈ (Met‘𝑋) → dom 𝐷 = (𝑋 × 𝑋))
76eqeq1d 2653 . . . 4 (𝐷 ∈ (Met‘𝑋) → (dom 𝐷 = ∅ ↔ (𝑋 × 𝑋) = ∅))
84, 7bitrd 268 . . 3 (𝐷 ∈ (Met‘𝑋) → (𝐷 = ∅ ↔ (𝑋 × 𝑋) = ∅))
9 xpeq0 5589 . . . 4 ((𝑋 × 𝑋) = ∅ ↔ (𝑋 = ∅ ∨ 𝑋 = ∅))
10 oridm 535 . . . 4 ((𝑋 = ∅ ∨ 𝑋 = ∅) ↔ 𝑋 = ∅)
119, 10bitri 264 . . 3 ((𝑋 × 𝑋) = ∅ ↔ 𝑋 = ∅)
128, 11syl6bb 276 . 2 (𝐷 ∈ (Met‘𝑋) → (𝐷 = ∅ ↔ 𝑋 = ∅))
1312necon3bid 2867 1 (𝐷 ∈ (Met‘𝑋) → (𝐷 ≠ ∅ ↔ 𝑋 ≠ ∅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 382   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  ∅c0 3948   × cxp 5141  dom cdm 5143  Rel wrel 5148  ⟶wf 5922  ‘cfv 5926  ℝcr 9973  Metcme 19780 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-met 19788 This theorem is referenced by: (None)
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