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Mirrors > Home > ILE Home > Th. List > posng | Unicode version |
Description: Partial ordering of a singleton. (Contributed by Jim Kingdon, 5-Dec-2018.) |
Ref | Expression |
---|---|
posng |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-po 4090 |
. 2
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2 | breq2 3818 |
. . . . . . . . . . 11
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3 | 2 | anbi2d 452 |
. . . . . . . . . 10
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4 | breq2 3818 |
. . . . . . . . . 10
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5 | 3, 4 | imbi12d 232 |
. . . . . . . . 9
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6 | 5 | anbi2d 452 |
. . . . . . . 8
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7 | 6 | ralsng 3460 |
. . . . . . 7
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8 | 7 | ralbidv 2376 |
. . . . . 6
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9 | simpl 107 |
. . . . . . . . . 10
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10 | breq2 3818 |
. . . . . . . . . 10
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11 | 9, 10 | syl5ib 152 |
. . . . . . . . 9
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12 | 11 | biantrud 298 |
. . . . . . . 8
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13 | 12 | bicomd 139 |
. . . . . . 7
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14 | 13 | ralsng 3460 |
. . . . . 6
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15 | 8, 14 | bitrd 186 |
. . . . 5
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16 | 15 | ralbidv 2376 |
. . . 4
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17 | breq12 3819 |
. . . . . . 7
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18 | 17 | anidms 389 |
. . . . . 6
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19 | 18 | notbid 625 |
. . . . 5
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20 | 19 | ralsng 3460 |
. . . 4
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21 | 16, 20 | bitrd 186 |
. . 3
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22 | 21 | adantl 271 |
. 2
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23 | 1, 22 | syl5bb 190 |
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 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 |
This theorem depends on definitions: df-bi 115 df-3an 924 df-tru 1290 df-nf 1393 df-sb 1690 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-ral 2360 df-v 2616 df-sbc 2829 df-un 2990 df-sn 3431 df-pr 3432 df-op 3434 df-br 3815 df-po 4090 |
This theorem is referenced by: sosng 4472 |
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