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Theorem metn0 22965
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 22935 . . . . 5 (𝐷 ∈ (Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ)
2 frel 6499 . . . . 5 (𝐷:(𝑋 × 𝑋)⟶ℝ → Rel 𝐷)
3 reldm0 5775 . . . . 5 (Rel 𝐷 → (𝐷 = ∅ ↔ dom 𝐷 = ∅))
41, 2, 33syl 18 . . . 4 (𝐷 ∈ (Met‘𝑋) → (𝐷 = ∅ ↔ dom 𝐷 = ∅))
51fdmd 6504 . . . . 5 (𝐷 ∈ (Met‘𝑋) → dom 𝐷 = (𝑋 × 𝑋))
65eqeq1d 2824 . . . 4 (𝐷 ∈ (Met‘𝑋) → (dom 𝐷 = ∅ ↔ (𝑋 × 𝑋) = ∅))
74, 6bitrd 282 . . 3 (𝐷 ∈ (Met‘𝑋) → (𝐷 = ∅ ↔ (𝑋 × 𝑋) = ∅))
8 xpeq0 5995 . . . 4 ((𝑋 × 𝑋) = ∅ ↔ (𝑋 = ∅ ∨ 𝑋 = ∅))
9 oridm 902 . . . 4 ((𝑋 = ∅ ∨ 𝑋 = ∅) ↔ 𝑋 = ∅)
108, 9bitri 278 . . 3 ((𝑋 × 𝑋) = ∅ ↔ 𝑋 = ∅)
117, 10syl6bb 290 . 2 (𝐷 ∈ (Met‘𝑋) → (𝐷 = ∅ ↔ 𝑋 = ∅))
1211necon3bid 3055 1 (𝐷 ∈ (Met‘𝑋) → (𝐷 ≠ ∅ ↔ 𝑋 ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wo 844   = wceq 1538  wcel 2114  wne 3011  c0 4265   × cxp 5530  dom cdm 5532  Rel wrel 5537  wf 6330  cfv 6334  cr 10525  Metcmet 20075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-resscn 10583
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-map 8395  df-met 20083
This theorem is referenced by: (None)
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