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Mirrors > Home > ILE Home > Th. List > tridc | Unicode version |
Description: A trichotomous order is decidable. (Contributed by Jim Kingdon, 5-Sep-2022.) |
Ref | Expression |
---|---|
tridc.po | |
tridc.tri | |
tridc.b | |
tridc.c |
Ref | Expression |
---|---|
tridc | DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 109 | . . . 4 | |
2 | 1 | orcd 722 | . . 3 |
3 | df-dc 820 | . . 3 DECID | |
4 | 2, 3 | sylibr 133 | . 2 DECID |
5 | tridc.po | . . . . . . 7 | |
6 | tridc.c | . . . . . . 7 | |
7 | poirr 4224 | . . . . . . 7 | |
8 | 5, 6, 7 | syl2anc 408 | . . . . . 6 |
9 | 8 | adantr 274 | . . . . 5 |
10 | simpr 109 | . . . . . 6 | |
11 | 10 | breq1d 3934 | . . . . 5 |
12 | 9, 11 | mtbird 662 | . . . 4 |
13 | 12 | olcd 723 | . . 3 |
14 | 13, 3 | sylibr 133 | . 2 DECID |
15 | tridc.b | . . . . . . 7 | |
16 | po2nr 4226 | . . . . . . 7 | |
17 | 5, 6, 15, 16 | syl12anc 1214 | . . . . . 6 |
18 | 17 | adantr 274 | . . . . 5 |
19 | simplr 519 | . . . . . 6 | |
20 | simpr 109 | . . . . . 6 | |
21 | 19, 20 | jca 304 | . . . . 5 |
22 | 18, 21 | mtand 654 | . . . 4 |
23 | 22 | olcd 723 | . . 3 |
24 | 23, 3 | sylibr 133 | . 2 DECID |
25 | tridc.tri | . . 3 | |
26 | breq1 3927 | . . . . 5 | |
27 | eqeq1 2144 | . . . . 5 | |
28 | breq2 3928 | . . . . 5 | |
29 | 26, 27, 28 | 3orbi123d 1289 | . . . 4 |
30 | breq2 3928 | . . . . 5 | |
31 | eqeq2 2147 | . . . . 5 | |
32 | breq1 3927 | . . . . 5 | |
33 | 30, 31, 32 | 3orbi123d 1289 | . . . 4 |
34 | 29, 33 | rspc2va 2798 | . . 3 |
35 | 15, 6, 25, 34 | syl21anc 1215 | . 2 |
36 | 4, 14, 24, 35 | mpjao3dan 1285 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 697 DECID wdc 819 w3o 961 wceq 1331 wcel 1480 wral 2414 class class class wbr 3924 wpo 4211 |
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-dc 820 df-3or 963 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-un 3070 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-po 4213 |
This theorem is referenced by: fimax2gtrilemstep 6787 |
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