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Mirrors > Home > ILE Home > Th. List > sotricim | Unicode version |
Description: One direction of sotritric 4162 holds for all weakly linear orders. (Contributed by Jim Kingdon, 28-Sep-2019.) |
Ref | Expression |
---|---|
sotricim |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sonr 4155 |
. . . . . . 7
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2 | 1 | adantrr 464 |
. . . . . 6
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3 | 2 | 3adant3 964 |
. . . . 5
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4 | breq2 3857 |
. . . . . . 7
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5 | 4 | biimprcd 159 |
. . . . . 6
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6 | 5 | 3ad2ant3 967 |
. . . . 5
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7 | 3, 6 | mtod 625 |
. . . 4
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8 | 7 | 3expia 1146 |
. . 3
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9 | so2nr 4159 |
. . . 4
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10 | imnan 660 |
. . . 4
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11 | 9, 10 | sylibr 133 |
. . 3
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12 | 8, 11 | jcad 302 |
. 2
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13 | ioran 705 |
. 2
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14 | 12, 13 | syl6ibr 161 |
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-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 2624 df-un 3006 df-sn 3458 df-pr 3459 df-op 3461 df-br 3854 df-po 4134 df-iso 4135 |
This theorem is referenced by: sotritric 4162 |
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