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Mirrors > Home > MPE Home > Th. List > metn0 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
metn0 | ⊢ (𝐷 ∈ (Met‘𝑋) → (𝐷 ≠ ∅ ↔ 𝑋 ≠ ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metf 22934 | . . . . 5 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ) | |
2 | frel 6514 | . . . . 5 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ → Rel 𝐷) | |
3 | reldm0 5793 | . . . . 5 ⊢ (Rel 𝐷 → (𝐷 = ∅ ↔ dom 𝐷 = ∅)) | |
4 | 1, 2, 3 | 3syl 18 | . . . 4 ⊢ (𝐷 ∈ (Met‘𝑋) → (𝐷 = ∅ ↔ dom 𝐷 = ∅)) |
5 | 1 | fdmd 6518 | . . . . 5 ⊢ (𝐷 ∈ (Met‘𝑋) → dom 𝐷 = (𝑋 × 𝑋)) |
6 | 5 | eqeq1d 2823 | . . . 4 ⊢ (𝐷 ∈ (Met‘𝑋) → (dom 𝐷 = ∅ ↔ (𝑋 × 𝑋) = ∅)) |
7 | 4, 6 | bitrd 281 | . . 3 ⊢ (𝐷 ∈ (Met‘𝑋) → (𝐷 = ∅ ↔ (𝑋 × 𝑋) = ∅)) |
8 | xpeq0 6012 | . . . 4 ⊢ ((𝑋 × 𝑋) = ∅ ↔ (𝑋 = ∅ ∨ 𝑋 = ∅)) | |
9 | oridm 901 | . . . 4 ⊢ ((𝑋 = ∅ ∨ 𝑋 = ∅) ↔ 𝑋 = ∅) | |
10 | 8, 9 | bitri 277 | . . 3 ⊢ ((𝑋 × 𝑋) = ∅ ↔ 𝑋 = ∅) |
11 | 7, 10 | syl6bb 289 | . 2 ⊢ (𝐷 ∈ (Met‘𝑋) → (𝐷 = ∅ ↔ 𝑋 = ∅)) |
12 | 11 | necon3bid 3060 | 1 ⊢ (𝐷 ∈ (Met‘𝑋) → (𝐷 ≠ ∅ ↔ 𝑋 ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∨ wo 843 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∅c0 4291 × cxp 5548 dom cdm 5550 Rel wrel 5555 ⟶wf 6346 ‘cfv 6350 ℝcr 10530 Metcmet 20525 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-map 8402 df-met 20533 |
This theorem is referenced by: (None) |
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