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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 22943 | . . . . 5 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ) | |
2 | frel 6522 | . . . . 5 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ → Rel 𝐷) | |
3 | reldm0 5801 | . . . . 5 ⊢ (Rel 𝐷 → (𝐷 = ∅ ↔ dom 𝐷 = ∅)) | |
4 | 1, 2, 3 | 3syl 18 | . . . 4 ⊢ (𝐷 ∈ (Met‘𝑋) → (𝐷 = ∅ ↔ dom 𝐷 = ∅)) |
5 | 1 | fdmd 6526 | . . . . 5 ⊢ (𝐷 ∈ (Met‘𝑋) → dom 𝐷 = (𝑋 × 𝑋)) |
6 | 5 | eqeq1d 2826 | . . . 4 ⊢ (𝐷 ∈ (Met‘𝑋) → (dom 𝐷 = ∅ ↔ (𝑋 × 𝑋) = ∅)) |
7 | 4, 6 | bitrd 281 | . . 3 ⊢ (𝐷 ∈ (Met‘𝑋) → (𝐷 = ∅ ↔ (𝑋 × 𝑋) = ∅)) |
8 | xpeq0 6020 | . . . 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 3063 | 1 ⊢ (𝐷 ∈ (Met‘𝑋) → (𝐷 ≠ ∅ ↔ 𝑋 ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∨ wo 843 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ∅c0 4294 × cxp 5556 dom cdm 5558 Rel wrel 5563 ⟶wf 6354 ‘cfv 6358 ℝcr 10539 Metcmet 20534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-map 8411 df-met 20542 |
This theorem is referenced by: (None) |
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