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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 4130 | . 2 ⊢ ∅ ∈ V | |
2 | f0 5406 | . . 3 ⊢ ∅:∅⟶ℝ | |
3 | xp0 5048 | . . . 4 ⊢ (∅ × ∅) = ∅ | |
4 | 3 | feq2i 5359 | . . 3 ⊢ (∅:(∅ × ∅)⟶ℝ ↔ ∅:∅⟶ℝ) |
5 | 2, 4 | mpbir 146 | . 2 ⊢ ∅:(∅ × ∅)⟶ℝ |
6 | noel 3426 | . . . 4 ⊢ ¬ 𝑥 ∈ ∅ | |
7 | 6 | pm2.21i 646 | . . 3 ⊢ (𝑥 ∈ ∅ → ((𝑥∅𝑦) = 0 ↔ 𝑥 = 𝑦)) |
8 | 7 | adantr 276 | . 2 ⊢ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅) → ((𝑥∅𝑦) = 0 ↔ 𝑥 = 𝑦)) |
9 | 6 | pm2.21i 646 | . . 3 ⊢ (𝑥 ∈ ∅ → (𝑥∅𝑦) ≤ ((𝑧∅𝑥) + (𝑧∅𝑦))) |
10 | 9 | 3ad2ant1 1018 | . 2 ⊢ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) → (𝑥∅𝑦) ≤ ((𝑧∅𝑥) + (𝑧∅𝑦))) |
11 | 1, 5, 8, 10 | ismeti 13777 | 1 ⊢ ∅ ∈ (Met‘∅) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 = wceq 1353 ∈ wcel 2148 ∅c0 3422 class class class wbr 4003 × cxp 4624 ⟶wf 5212 ‘cfv 5216 (class class class)co 5874 ℝcr 7809 0cc0 7810 + caddc 7813 ≤ cle 7991 Metcmet 13372 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-fv 5224 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-map 6649 df-met 13380 |
This theorem is referenced by: (None) |
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