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Theorem metn0 22386
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 22356 . . . . 5 (𝐷 ∈ (Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ)
2 frel 6191 . . . . 5 (𝐷:(𝑋 × 𝑋)⟶ℝ → Rel 𝐷)
3 reldm0 5482 . . . . 5 (Rel 𝐷 → (𝐷 = ∅ ↔ dom 𝐷 = ∅))
41, 2, 33syl 18 . . . 4 (𝐷 ∈ (Met‘𝑋) → (𝐷 = ∅ ↔ dom 𝐷 = ∅))
5 fdm 6192 . . . . . 6 (𝐷:(𝑋 × 𝑋)⟶ℝ → dom 𝐷 = (𝑋 × 𝑋))
61, 5syl 17 . . . . 5 (𝐷 ∈ (Met‘𝑋) → dom 𝐷 = (𝑋 × 𝑋))
76eqeq1d 2773 . . . 4 (𝐷 ∈ (Met‘𝑋) → (dom 𝐷 = ∅ ↔ (𝑋 × 𝑋) = ∅))
84, 7bitrd 268 . . 3 (𝐷 ∈ (Met‘𝑋) → (𝐷 = ∅ ↔ (𝑋 × 𝑋) = ∅))
9 xpeq0 5696 . . . 4 ((𝑋 × 𝑋) = ∅ ↔ (𝑋 = ∅ ∨ 𝑋 = ∅))
10 oridm 882 . . . 4 ((𝑋 = ∅ ∨ 𝑋 = ∅) ↔ 𝑋 = ∅)
119, 10bitri 264 . . 3 ((𝑋 × 𝑋) = ∅ ↔ 𝑋 = ∅)
128, 11syl6bb 276 . 2 (𝐷 ∈ (Met‘𝑋) → (𝐷 = ∅ ↔ 𝑋 = ∅))
1312necon3bid 2987 1 (𝐷 ∈ (Met‘𝑋) → (𝐷 ≠ ∅ ↔ 𝑋 ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 828   = wceq 1631  wcel 2145  wne 2943  c0 4064   × cxp 5248  dom cdm 5250  Rel wrel 5255  wf 6028  cfv 6032  cr 10138  Metcme 19948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097  ax-cnex 10195  ax-resscn 10196
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3589  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-fv 6040  df-ov 6797  df-oprab 6798  df-mpt2 6799  df-map 8012  df-met 19956
This theorem is referenced by: (None)
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