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Mirrors > Home > ILE Home > Th. List > ordtriexmidlem | Unicode version |
Description: Lemma for decidability and ordinals. The set is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4482 or weak linearity in ordsoexmid 4523) with a proposition . Our lemma states that it is an ordinal number. (Contributed by Jim Kingdon, 28-Jan-2019.) |
Ref | Expression |
---|---|
ordtriexmidlem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 | . . . . . 6 | |
2 | elrabi 2865 | . . . . . . . . 9 | |
3 | velsn 3578 | . . . . . . . . 9 | |
4 | 2, 3 | sylib 121 | . . . . . . . 8 |
5 | noel 3399 | . . . . . . . . 9 | |
6 | eleq2 2221 | . . . . . . . . 9 | |
7 | 5, 6 | mtbiri 665 | . . . . . . . 8 |
8 | 4, 7 | syl 14 | . . . . . . 7 |
9 | 8 | adantl 275 | . . . . . 6 |
10 | 1, 9 | pm2.21dd 610 | . . . . 5 |
11 | 10 | gen2 1430 | . . . 4 |
12 | dftr2 4066 | . . . 4 | |
13 | 11, 12 | mpbir 145 | . . 3 |
14 | ssrab2 3213 | . . 3 | |
15 | ord0 4353 | . . . . 5 | |
16 | ordsucim 4461 | . . . . 5 | |
17 | 15, 16 | ax-mp 5 | . . . 4 |
18 | suc0 4373 | . . . . 5 | |
19 | ordeq 4334 | . . . . 5 | |
20 | 18, 19 | ax-mp 5 | . . . 4 |
21 | 17, 20 | mpbi 144 | . . 3 |
22 | trssord 4342 | . . 3 | |
23 | 13, 14, 21, 22 | mp3an 1319 | . 2 |
24 | p0ex 4151 | . . . 4 | |
25 | 24 | rabex 4110 | . . 3 |
26 | 25 | elon 4336 | . 2 |
27 | 23, 26 | mpbir 145 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wal 1333 wceq 1335 wcel 2128 crab 2439 wss 3102 c0 3395 csn 3561 wtr 4064 word 4324 con0 4325 csuc 4327 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-nul 4092 ax-pow 4137 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3396 df-pw 3546 df-sn 3567 df-uni 3775 df-tr 4065 df-iord 4328 df-on 4330 df-suc 4333 |
This theorem is referenced by: ordtriexmid 4482 ontriexmidim 4483 ordtri2orexmid 4484 ontr2exmid 4486 onsucsssucexmid 4488 ordsoexmid 4523 0elsucexmid 4526 ordpwsucexmid 4531 unfiexmid 6864 exmidonfinlem 7130 |
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