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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 4063 | . 2 ⊢ ∅ ∈ V | |
2 | f0 5321 | . . 3 ⊢ ∅:∅⟶ℝ | |
3 | xp0 4966 | . . . 4 ⊢ (∅ × ∅) = ∅ | |
4 | 3 | feq2i 5274 | . . 3 ⊢ (∅:(∅ × ∅)⟶ℝ ↔ ∅:∅⟶ℝ) |
5 | 2, 4 | mpbir 145 | . 2 ⊢ ∅:(∅ × ∅)⟶ℝ |
6 | noel 3372 | . . . 4 ⊢ ¬ 𝑥 ∈ ∅ | |
7 | 6 | pm2.21i 636 | . . 3 ⊢ (𝑥 ∈ ∅ → ((𝑥∅𝑦) = 0 ↔ 𝑥 = 𝑦)) |
8 | 7 | adantr 274 | . 2 ⊢ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅) → ((𝑥∅𝑦) = 0 ↔ 𝑥 = 𝑦)) |
9 | 6 | pm2.21i 636 | . . 3 ⊢ (𝑥 ∈ ∅ → (𝑥∅𝑦) ≤ ((𝑧∅𝑥) + (𝑧∅𝑦))) |
10 | 9 | 3ad2ant1 1003 | . 2 ⊢ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) → (𝑥∅𝑦) ≤ ((𝑧∅𝑥) + (𝑧∅𝑦))) |
11 | 1, 5, 8, 10 | ismeti 12554 | 1 ⊢ ∅ ∈ (Met‘∅) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1332 ∈ wcel 1481 ∅c0 3368 class class class wbr 3937 × cxp 4545 ⟶wf 5127 ‘cfv 5131 (class class class)co 5782 ℝcr 7643 0cc0 7644 + caddc 7647 ≤ cle 7825 Metcmet 12189 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-map 6552 df-met 12197 |
This theorem is referenced by: (None) |
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