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Mirrors > Home > MPE Home > Th. List > metn0 | Structured version Visualization version GIF version |
Description: A metric space is nonempty iff its base set is nonempty. (Contributed by NM, 4-Oct-2007.) (Revised by Mario Carneiro, 14-Aug-2015.) |
Ref | Expression |
---|---|
metn0 | β’ (π· β (Metβπ) β (π· β β β π β β )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metf 24235 | . . . . 5 β’ (π· β (Metβπ) β π·:(π Γ π)βΆβ) | |
2 | frel 6727 | . . . . 5 β’ (π·:(π Γ π)βΆβ β Rel π·) | |
3 | reldm0 5930 | . . . . 5 β’ (Rel π· β (π· = β β dom π· = β )) | |
4 | 1, 2, 3 | 3syl 18 | . . . 4 β’ (π· β (Metβπ) β (π· = β β dom π· = β )) |
5 | 1 | fdmd 6733 | . . . . 5 β’ (π· β (Metβπ) β dom π· = (π Γ π)) |
6 | 5 | eqeq1d 2730 | . . . 4 β’ (π· β (Metβπ) β (dom π· = β β (π Γ π) = β )) |
7 | 4, 6 | bitrd 279 | . . 3 β’ (π· β (Metβπ) β (π· = β β (π Γ π) = β )) |
8 | xpeq0 6164 | . . . 4 β’ ((π Γ π) = β β (π = β β¨ π = β )) | |
9 | oridm 903 | . . . 4 β’ ((π = β β¨ π = β ) β π = β ) | |
10 | 8, 9 | bitri 275 | . . 3 β’ ((π Γ π) = β β π = β ) |
11 | 7, 10 | bitrdi 287 | . 2 β’ (π· β (Metβπ) β (π· = β β π = β )) |
12 | 11 | necon3bid 2982 | 1 β’ (π· β (Metβπ) β (π· β β β π β β )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β¨ wo 846 = wceq 1534 β wcel 2099 β wne 2937 β c0 4323 Γ cxp 5676 dom cdm 5678 Rel wrel 5683 βΆwf 6544 βcfv 6548 βcr 11137 Metcmet 21264 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-map 8846 df-met 21272 |
This theorem is referenced by: (None) |
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