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Mirrors > Home > MPE Home > Th. List > metn0 | Unicode 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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metf 18321 |
. . . . 5
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2 | frel 5561 |
. . . . 5
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3 | reldm0 5054 |
. . . . 5
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4 | 1, 2, 3 | 3syl 19 |
. . . 4
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5 | fdm 5562 |
. . . . . 6
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6 | 1, 5 | syl 16 |
. . . . 5
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7 | 6 | eqeq1d 2420 |
. . . 4
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8 | 4, 7 | bitrd 245 |
. . 3
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9 | xpeq0 5260 |
. . . 4
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10 | oridm 501 |
. . . 4
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11 | 9, 10 | bitri 241 |
. . 3
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12 | 8, 11 | syl6bb 253 |
. 2
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13 | 12 | necon3bid 2610 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1552 ax-5 1563 ax-17 1623 ax-9 1662 ax-8 1683 ax-13 1723 ax-14 1725 ax-6 1740 ax-7 1745 ax-11 1757 ax-12 1946 ax-ext 2393 ax-sep 4298 ax-nul 4306 ax-pow 4345 ax-pr 4371 ax-un 4668 ax-cnex 9010 ax-resscn 9011 |
This theorem depends on definitions: df-bi 178 df-or 360 df-an 361 df-3an 938 df-tru 1325 df-ex 1548 df-nf 1551 df-sb 1656 df-eu 2266 df-mo 2267 df-clab 2399 df-cleq 2405 df-clel 2408 df-nfc 2537 df-ne 2577 df-ral 2679 df-rex 2680 df-rab 2683 df-v 2926 df-sbc 3130 df-dif 3291 df-un 3293 df-in 3295 df-ss 3302 df-nul 3597 df-if 3708 df-pw 3769 df-sn 3788 df-pr 3789 df-op 3791 df-uni 3984 df-br 4181 df-opab 4235 df-mpt 4236 df-id 4466 df-xp 4851 df-rel 4852 df-cnv 4853 df-co 4854 df-dm 4855 df-rn 4856 df-iota 5385 df-fun 5423 df-fn 5424 df-f 5425 df-fv 5429 df-ov 6051 df-oprab 6052 df-mpt2 6053 df-map 6987 df-met 16659 |
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