Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 4092 | . 2 ⊢ ∅ ∈ V | |
2 | f0 5361 | . . 3 ⊢ ∅:∅⟶ℝ | |
3 | xp0 5006 | . . . 4 ⊢ (∅ × ∅) = ∅ | |
4 | 3 | feq2i 5314 | . . 3 ⊢ (∅:(∅ × ∅)⟶ℝ ↔ ∅:∅⟶ℝ) |
5 | 2, 4 | mpbir 145 | . 2 ⊢ ∅:(∅ × ∅)⟶ℝ |
6 | noel 3398 | . . . 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 12788 | 1 ⊢ ∅ ∈ (Met‘∅) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1335 ∈ wcel 2128 ∅c0 3394 class class class wbr 3966 × cxp 4585 ⟶wf 5167 ‘cfv 5171 (class class class)co 5825 ℝcr 7732 0cc0 7733 + caddc 7736 ≤ cle 7914 Metcmet 12423 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-nul 4091 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-cnex 7824 ax-resscn 7825 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-iun 3852 df-br 3967 df-opab 4027 df-mpt 4028 df-id 4254 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-fv 5179 df-ov 5828 df-oprab 5829 df-mpo 5830 df-1st 6089 df-2nd 6090 df-map 6596 df-met 12431 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |