Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > sotric | Structured version Visualization version Unicode version |
Description: A strict order relation satisfies strict trichotomy. (Contributed by NM, 19-Feb-1996.) |
Ref | Expression |
---|---|
sotric |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sonr 5056 | . . . . . 6 | |
2 | breq2 4657 | . . . . . . 7 | |
3 | 2 | notbid 308 | . . . . . 6 |
4 | 1, 3 | syl5ibcom 235 | . . . . 5 |
5 | 4 | adantrr 753 | . . . 4 |
6 | so2nr 5059 | . . . . . 6 | |
7 | imnan 438 | . . . . . 6 | |
8 | 6, 7 | sylibr 224 | . . . . 5 |
9 | 8 | con2d 129 | . . . 4 |
10 | 5, 9 | jaod 395 | . . 3 |
11 | solin 5058 | . . . . 5 | |
12 | 3orass 1040 | . . . . 5 | |
13 | 11, 12 | sylib 208 | . . . 4 |
14 | 13 | ord 392 | . . 3 |
15 | 10, 14 | impbid 202 | . 2 |
16 | 15 | con2bid 344 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wo 383 wa 384 w3o 1036 wceq 1483 wcel 1990 class class class wbr 4653 wor 5034 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-po 5035 df-so 5036 |
This theorem is referenced by: sotr2 5064 sotri2 5525 sotri3 5526 somin1 5529 somincom 5530 soisores 6577 soisoi 6578 fimaxg 8207 suplub2 8367 supgtoreq 8376 fiming 8404 ordtypelem7 8429 fpwwe2 9465 indpi 9729 nqereu 9751 ltsonq 9791 prub 9816 ltapr 9867 suplem2pr 9875 ltsosr 9915 axpre-lttri 9986 sotr3 31656 soasym 31657 noetalem3 31865 sleloe 31879 |
Copyright terms: Public domain | W3C validator |