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Mirrors > Home > MPE Home > Th. List > xmet0 | Structured version Visualization version GIF version |
Description: The distance function of a metric space is zero if its arguments are equal. Definition 14-1.1(a) of [Gleason] p. 223. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xmet0 | ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2735 | . 2 ⊢ 𝐴 = 𝐴 | |
2 | xmeteq0 24364 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐷𝐴) = 0 ↔ 𝐴 = 𝐴)) | |
3 | 2 | 3anidm23 1420 | . 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐷𝐴) = 0 ↔ 𝐴 = 𝐴)) |
4 | 1, 3 | mpbiri 258 | 1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 0cc0 11153 ∞Metcxmet 21367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8867 df-xr 11297 df-xmet 21375 |
This theorem is referenced by: met0 24369 xmetge0 24370 xmetsym 24373 xmetpsmet 24374 xblcntr 24437 ssbl 24449 xmeter 24459 ubthlem2 30900 sitmcl 34333 |
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