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Mirrors > Home > ILE Home > Th. List > 0met | 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 4116 | . 2 ⊢ ∅ ∈ V | |
2 | f0 5388 | . . 3 ⊢ ∅:∅⟶ℝ | |
3 | xp0 5030 | . . . 4 ⊢ (∅ × ∅) = ∅ | |
4 | 3 | feq2i 5341 | . . 3 ⊢ (∅:(∅ × ∅)⟶ℝ ↔ ∅:∅⟶ℝ) |
5 | 2, 4 | mpbir 145 | . 2 ⊢ ∅:(∅ × ∅)⟶ℝ |
6 | noel 3418 | . . . 4 ⊢ ¬ 𝑥 ∈ ∅ | |
7 | 6 | pm2.21i 641 | . . 3 ⊢ (𝑥 ∈ ∅ → ((𝑥∅𝑦) = 0 ↔ 𝑥 = 𝑦)) |
8 | 7 | adantr 274 | . 2 ⊢ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅) → ((𝑥∅𝑦) = 0 ↔ 𝑥 = 𝑦)) |
9 | 6 | pm2.21i 641 | . . 3 ⊢ (𝑥 ∈ ∅ → (𝑥∅𝑦) ≤ ((𝑧∅𝑥) + (𝑧∅𝑦))) |
10 | 9 | 3ad2ant1 1013 | . 2 ⊢ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) → (𝑥∅𝑦) ≤ ((𝑧∅𝑥) + (𝑧∅𝑦))) |
11 | 1, 5, 8, 10 | ismeti 13140 | 1 ⊢ ∅ ∈ (Met‘∅) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1348 ∈ wcel 2141 ∅c0 3414 class class class wbr 3989 × cxp 4609 ⟶wf 5194 ‘cfv 5198 (class class class)co 5853 ℝcr 7773 0cc0 7774 + caddc 7777 ≤ cle 7955 Metcmet 12775 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-map 6628 df-met 12783 |
This theorem is referenced by: (None) |
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