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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 4281 | . 2 | |
2 | breq2 3993 | . . . . . . . . . . 11 | |
3 | 2 | anbi2d 461 | . . . . . . . . . 10 |
4 | breq2 3993 | . . . . . . . . . 10 | |
5 | 3, 4 | imbi12d 233 | . . . . . . . . 9 |
6 | 5 | anbi2d 461 | . . . . . . . 8 |
7 | 6 | ralsng 3623 | . . . . . . 7 |
8 | 7 | ralbidv 2470 | . . . . . 6 |
9 | simpl 108 | . . . . . . . . . 10 | |
10 | breq2 3993 | . . . . . . . . . 10 | |
11 | 9, 10 | syl5ib 153 | . . . . . . . . 9 |
12 | 11 | biantrud 302 | . . . . . . . 8 |
13 | 12 | bicomd 140 | . . . . . . 7 |
14 | 13 | ralsng 3623 | . . . . . 6 |
15 | 8, 14 | bitrd 187 | . . . . 5 |
16 | 15 | ralbidv 2470 | . . . 4 |
17 | breq12 3994 | . . . . . . 7 | |
18 | 17 | anidms 395 | . . . . . 6 |
19 | 18 | notbid 662 | . . . . 5 |
20 | 19 | ralsng 3623 | . . . 4 |
21 | 16, 20 | bitrd 187 | . . 3 |
22 | 21 | adantl 275 | . 2 |
23 | 1, 22 | syl5bb 191 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1348 wcel 2141 wral 2448 cvv 2730 csn 3583 class class class wbr 3989 wpo 4279 wrel 4616 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-v 2732 df-sbc 2956 df-un 3125 df-sn 3589 df-pr 3590 df-op 3592 df-br 3990 df-po 4281 |
This theorem is referenced by: sosng 4684 |
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