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Mirrors > Home > ILE Home > Th. List > sotritrieq | Unicode version |
Description: A trichotomy relationship, given a trichotomous order. (Contributed by Jim Kingdon, 13-Dec-2019.) |
Ref | Expression |
---|---|
sotritric.or |
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sotritric.tri |
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Ref | Expression |
---|---|
sotritrieq |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sotritric.or |
. . . . . . 7
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2 | sonr 4153 |
. . . . . . 7
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3 | 1, 2 | mpan 416 |
. . . . . 6
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4 | breq2 3855 |
. . . . . . 7
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5 | 4 | notbid 628 |
. . . . . 6
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6 | 3, 5 | syl5ibcom 154 |
. . . . 5
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7 | breq1 3854 |
. . . . . . 7
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8 | 7 | notbid 628 |
. . . . . 6
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9 | 3, 8 | syl5ibcom 154 |
. . . . 5
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10 | 6, 9 | jcad 302 |
. . . 4
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11 | ioran 705 |
. . . 4
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12 | 10, 11 | syl6ibr 161 |
. . 3
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13 | 12 | adantr 271 |
. 2
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14 | sotritric.tri |
. . 3
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15 | 3orrot 931 |
. . . . . . 7
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16 | 3orcomb 934 |
. . . . . . 7
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17 | 3orass 928 |
. . . . . . 7
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18 | 15, 16, 17 | 3bitri 205 |
. . . . . 6
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19 | 18 | biimpi 119 |
. . . . 5
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20 | 19 | orcomd 684 |
. . . 4
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21 | 20 | ord 679 |
. . 3
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22 | 14, 21 | syl 14 |
. 2
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23 | 13, 22 | impbid 128 |
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-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-3or 926 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-v 2622 df-un 3004 df-sn 3456 df-pr 3457 df-op 3459 df-br 3852 df-po 4132 df-iso 4133 |
This theorem is referenced by: distrlem4prl 7204 distrlem4pru 7205 |
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