Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 4213 | . 2 | |
2 | breq2 3928 | . . . . . . . . . . 11 | |
3 | 2 | anbi2d 459 | . . . . . . . . . 10 |
4 | breq2 3928 | . . . . . . . . . 10 | |
5 | 3, 4 | imbi12d 233 | . . . . . . . . 9 |
6 | 5 | anbi2d 459 | . . . . . . . 8 |
7 | 6 | ralsng 3559 | . . . . . . 7 |
8 | 7 | ralbidv 2435 | . . . . . 6 |
9 | simpl 108 | . . . . . . . . . 10 | |
10 | breq2 3928 | . . . . . . . . . 10 | |
11 | 9, 10 | syl5ib 153 | . . . . . . . . 9 |
12 | 11 | biantrud 302 | . . . . . . . 8 |
13 | 12 | bicomd 140 | . . . . . . 7 |
14 | 13 | ralsng 3559 | . . . . . 6 |
15 | 8, 14 | bitrd 187 | . . . . 5 |
16 | 15 | ralbidv 2435 | . . . 4 |
17 | breq12 3929 | . . . . . . 7 | |
18 | 17 | anidms 394 | . . . . . 6 |
19 | 18 | notbid 656 | . . . . 5 |
20 | 19 | ralsng 3559 | . . . 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 1331 wcel 1480 wral 2414 cvv 2681 csn 3522 class class class wbr 3924 wpo 4211 wrel 4539 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-v 2683 df-sbc 2905 df-un 3070 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-po 4213 |
This theorem is referenced by: sosng 4607 |
Copyright terms: Public domain | W3C validator |