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Mirrors > Home > ILE Home > Th. List > sosng | Unicode version |
Description: Strict linear ordering on a singleton. (Contributed by Jim Kingdon, 5-Dec-2018.) |
Ref | Expression |
---|---|
sosng |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sopo 4205 | . . 3 | |
2 | posng 4581 | . . 3 | |
3 | 1, 2 | syl5ib 153 | . 2 |
4 | 2 | biimpar 295 | . . . 4 |
5 | ax-in2 589 | . . . . . . . . 9 | |
6 | 5 | adantr 274 | . . . . . . . 8 |
7 | elsni 3515 | . . . . . . . . . . 11 | |
8 | elsni 3515 | . . . . . . . . . . 11 | |
9 | 7, 8 | breqan12d 3915 | . . . . . . . . . 10 |
10 | 9 | imbi1d 230 | . . . . . . . . 9 |
11 | 10 | adantl 275 | . . . . . . . 8 |
12 | 6, 11 | mpbird 166 | . . . . . . 7 |
13 | 12 | ralrimivw 2483 | . . . . . 6 |
14 | 13 | ralrimivva 2491 | . . . . 5 |
15 | 14 | adantl 275 | . . . 4 |
16 | df-iso 4189 | . . . 4 | |
17 | 4, 15, 16 | sylanbrc 413 | . . 3 |
18 | 17 | ex 114 | . 2 |
19 | 3, 18 | impbid 128 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 682 wcel 1465 wral 2393 cvv 2660 csn 3497 class class class wbr 3899 wpo 4186 wor 4187 wrel 4514 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-v 2662 df-sbc 2883 df-un 3045 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-po 4188 df-iso 4189 |
This theorem is referenced by: (None) |
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