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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 4033 | . 2 | |
2 | breq2 3768 | . . . . . . . . . . 11 | |
3 | 2 | anbi2d 437 | . . . . . . . . . 10 |
4 | breq2 3768 | . . . . . . . . . 10 | |
5 | 3, 4 | imbi12d 223 | . . . . . . . . 9 |
6 | 5 | anbi2d 437 | . . . . . . . 8 |
7 | 6 | ralsng 3411 | . . . . . . 7 |
8 | 7 | ralbidv 2326 | . . . . . 6 |
9 | simpl 102 | . . . . . . . . . 10 | |
10 | breq2 3768 | . . . . . . . . . 10 | |
11 | 9, 10 | syl5ib 143 | . . . . . . . . 9 |
12 | 11 | biantrud 288 | . . . . . . . 8 |
13 | 12 | bicomd 129 | . . . . . . 7 |
14 | 13 | ralsng 3411 | . . . . . 6 |
15 | 8, 14 | bitrd 177 | . . . . 5 |
16 | 15 | ralbidv 2326 | . . . 4 |
17 | breq12 3769 | . . . . . . 7 | |
18 | 17 | anidms 377 | . . . . . 6 |
19 | 18 | notbid 592 | . . . . 5 |
20 | 19 | ralsng 3411 | . . . 4 |
21 | 16, 20 | bitrd 177 | . . 3 |
22 | 21 | adantl 262 | . 2 |
23 | 1, 22 | syl5bb 181 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wb 98 wceq 1243 wcel 1393 wral 2306 cvv 2557 csn 3375 class class class wbr 3764 wpo 4031 wrel 4350 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-v 2559 df-sbc 2765 df-un 2922 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-po 4033 |
This theorem is referenced by: sosng 4413 |
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