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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 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 = ∅)) | |
4 | 1, 2, 3 | 3syl 18 | . . . 4 ⊢ (𝐷 ∈ (Met‘𝑋) → (𝐷 = ∅ ↔ dom 𝐷 = ∅)) |
5 | 1 | fdmd 6733 | . . . . 5 ⊢ (𝐷 ∈ (Met‘𝑋) → dom 𝐷 = (𝑋 × 𝑋)) |
6 | 5 | eqeq1d 2727 | . . . 4 ⊢ (𝐷 ∈ (Met‘𝑋) → (dom 𝐷 = ∅ ↔ (𝑋 × 𝑋) = ∅)) |
7 | 4, 6 | bitrd 278 | . . 3 ⊢ (𝐷 ∈ (Met‘𝑋) → (𝐷 = ∅ ↔ (𝑋 × 𝑋) = ∅)) |
8 | xpeq0 6166 | . . . 4 ⊢ ((𝑋 × 𝑋) = ∅ ↔ (𝑋 = ∅ ∨ 𝑋 = ∅)) | |
9 | oridm 902 | . . . 4 ⊢ ((𝑋 = ∅ ∨ 𝑋 = ∅) ↔ 𝑋 = ∅) | |
10 | 8, 9 | bitri 274 | . . 3 ⊢ ((𝑋 × 𝑋) = ∅ ↔ 𝑋 = ∅) |
11 | 7, 10 | bitrdi 286 | . 2 ⊢ (𝐷 ∈ (Met‘𝑋) → (𝐷 = ∅ ↔ 𝑋 = ∅)) |
12 | 11 | necon3bid 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 ℝcr 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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