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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-po 4051 | . 2 | |
2 | breq2 3789 | . . . . . . . . . . 11 | |
3 | 2 | anbi2d 451 | . . . . . . . . . 10 |
4 | breq2 3789 | . . . . . . . . . 10 | |
5 | 3, 4 | imbi12d 232 | . . . . . . . . 9 |
6 | 5 | anbi2d 451 | . . . . . . . 8 |
7 | 6 | ralsng 3433 | . . . . . . 7 |
8 | 7 | ralbidv 2368 | . . . . . 6 |
9 | simpl 107 | . . . . . . . . . 10 | |
10 | breq2 3789 | . . . . . . . . . 10 | |
11 | 9, 10 | syl5ib 152 | . . . . . . . . 9 |
12 | 11 | biantrud 298 | . . . . . . . 8 |
13 | 12 | bicomd 139 | . . . . . . 7 |
14 | 13 | ralsng 3433 | . . . . . 6 |
15 | 8, 14 | bitrd 186 | . . . . 5 |
16 | 15 | ralbidv 2368 | . . . 4 |
17 | breq12 3790 | . . . . . . 7 | |
18 | 17 | anidms 389 | . . . . . 6 |
19 | 18 | notbid 624 | . . . . 5 |
20 | 19 | ralsng 3433 | . . . 4 |
21 | 16, 20 | bitrd 186 | . . 3 |
22 | 21 | adantl 271 | . 2 |
23 | 1, 22 | syl5bb 190 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 wceq 1284 wcel 1433 wral 2348 cvv 2601 csn 3398 class class class wbr 3785 wpo 4049 wrel 4368 |
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 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-v 2603 df-sbc 2816 df-un 2977 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-po 4051 |
This theorem is referenced by: sosng 4431 |
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