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Mirrors > Home > MPE Home > Th. List > 0met | Structured version Visualization version GIF version |
Description: The empty metric. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.) |
Ref | Expression |
---|---|
0met | ⊢ ∅ ∈ (Met‘∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5214 | . 2 ⊢ ∅ ∈ V | |
2 | f0 6563 | . . 3 ⊢ ∅:∅⟶ℝ | |
3 | xp0 6018 | . . . 4 ⊢ (∅ × ∅) = ∅ | |
4 | 3 | feq2i 6509 | . . 3 ⊢ (∅:(∅ × ∅)⟶ℝ ↔ ∅:∅⟶ℝ) |
5 | 2, 4 | mpbir 233 | . 2 ⊢ ∅:(∅ × ∅)⟶ℝ |
6 | noel 4299 | . . . 4 ⊢ ¬ 𝑥 ∈ ∅ | |
7 | 6 | pm2.21i 119 | . . 3 ⊢ (𝑥 ∈ ∅ → ((𝑥∅𝑦) = 0 ↔ 𝑥 = 𝑦)) |
8 | 7 | adantr 483 | . 2 ⊢ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅) → ((𝑥∅𝑦) = 0 ↔ 𝑥 = 𝑦)) |
9 | 6 | pm2.21i 119 | . . 3 ⊢ (𝑥 ∈ ∅ → (𝑥∅𝑦) ≤ ((𝑧∅𝑥) + (𝑧∅𝑦))) |
10 | 9 | 3ad2ant1 1129 | . 2 ⊢ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) → (𝑥∅𝑦) ≤ ((𝑧∅𝑥) + (𝑧∅𝑦))) |
11 | 1, 5, 8, 10 | ismeti 22938 | 1 ⊢ ∅ ∈ (Met‘∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1536 ∈ wcel 2113 ∅c0 4294 class class class wbr 5069 × cxp 5556 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 ℝcr 10539 0cc0 10540 + caddc 10543 ≤ cle 10679 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-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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