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Theorem metn0 24310
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 24280 . . . . 5 (𝐷 ∈ (Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ)
2 frel 6728 . . . . 5 (𝐷:(𝑋 × 𝑋)⟶ℝ → Rel 𝐷)
3 reldm0 5930 . . . . 5 (Rel 𝐷 → (𝐷 = ∅ ↔ dom 𝐷 = ∅))
41, 2, 33syl 18 . . . 4 (𝐷 ∈ (Met‘𝑋) → (𝐷 = ∅ ↔ dom 𝐷 = ∅))
51fdmd 6733 . . . . 5 (𝐷 ∈ (Met‘𝑋) → dom 𝐷 = (𝑋 × 𝑋))
65eqeq1d 2727 . . . 4 (𝐷 ∈ (Met‘𝑋) → (dom 𝐷 = ∅ ↔ (𝑋 × 𝑋) = ∅))
74, 6bitrd 278 . . 3 (𝐷 ∈ (Met‘𝑋) → (𝐷 = ∅ ↔ (𝑋 × 𝑋) = ∅))
8 xpeq0 6166 . . . 4 ((𝑋 × 𝑋) = ∅ ↔ (𝑋 = ∅ ∨ 𝑋 = ∅))
9 oridm 902 . . . 4 ((𝑋 = ∅ ∨ 𝑋 = ∅) ↔ 𝑋 = ∅)
108, 9bitri 274 . . 3 ((𝑋 × 𝑋) = ∅ ↔ 𝑋 = ∅)
117, 10bitrdi 286 . 2 (𝐷 ∈ (Met‘𝑋) → (𝐷 = ∅ ↔ 𝑋 = ∅))
1211necon3bid 2974 1 (𝐷 ∈ (Met‘𝑋) → (𝐷 ≠ ∅ ↔ 𝑋 ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wo 845   = wceq 1533  wcel 2098  wne 2929  c0 4322   × cxp 5676  dom cdm 5678  Rel wrel 5683  wf 6545  cfv 6549  cr 11139  Metcmet 21282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-map 8847  df-met 21290
This theorem is referenced by: (None)
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