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Mirrors > Home > MPE Home > Th. List > reldm0 | Structured version Visualization version Unicode version |
Description: A relation is empty iff its domain is empty. (Contributed by NM, 15-Sep-2004.) |
Ref | Expression |
---|---|
reldm0 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rel0 4961 |
. . 3
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2 | eqrel 4927 |
. . 3
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3 | 1, 2 | mpan2 678 |
. 2
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4 | eq0 3749 |
. . 3
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5 | alnex 1667 |
. . . . . 6
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6 | vex 3050 |
. . . . . . 7
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7 | 6 | eldm2 5036 |
. . . . . 6
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8 | 5, 7 | xchbinxr 313 |
. . . . 5
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9 | noel 3737 |
. . . . . . 7
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10 | 9 | nbn 349 |
. . . . . 6
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11 | 10 | albii 1693 |
. . . . 5
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12 | 8, 11 | bitr3i 255 |
. . . 4
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13 | 12 | albii 1693 |
. . 3
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14 | 4, 13 | bitr2i 254 |
. 2
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15 | 3, 14 | syl6bb 265 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1671 ax-4 1684 ax-5 1760 ax-6 1807 ax-7 1853 ax-9 1898 ax-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 ax-sep 4528 ax-nul 4537 ax-pr 4642 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 988 df-tru 1449 df-ex 1666 df-nf 1670 df-sb 1800 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2583 df-ne 2626 df-rab 2748 df-v 3049 df-dif 3409 df-un 3411 df-in 3413 df-ss 3420 df-nul 3734 df-if 3884 df-sn 3971 df-pr 3973 df-op 3977 df-br 4406 df-opab 4465 df-xp 4843 df-rel 4844 df-dm 4847 |
This theorem is referenced by: relrn0 5095 coeq0 5347 fnresdisj 5691 fn0 5700 fresaunres2 5760 fsnunfv 6109 frxp 6911 domss2 7736 swrd0 12797 setsres 15163 pmtrsn 17172 gsumval3 17553 00lsp 18216 metn0 21387 wlkn0 25267 usgravd00 25659 eupath 25721 dfrdg2 30454 mbfresfi 31999 mapfzcons1 35571 diophrw 35613 eldioph2lem1 35614 eldioph2lem2 35615 sge0cl 38233 funopsn 39028 1wlkn0 39644 |
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