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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 4505 or weak linearity in ordsoexmid 4546) 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 2883 | . . . . . . . . 9 | |
3 | velsn 3600 | . . . . . . . . 9 | |
4 | 2, 3 | sylib 121 | . . . . . . . 8 |
5 | noel 3418 | . . . . . . . . 9 | |
6 | eleq2 2234 | . . . . . . . . 9 | |
7 | 5, 6 | mtbiri 670 | . . . . . . . 8 |
8 | 4, 7 | syl 14 | . . . . . . 7 |
9 | 8 | adantl 275 | . . . . . 6 |
10 | 1, 9 | pm2.21dd 615 | . . . . 5 |
11 | 10 | gen2 1443 | . . . 4 |
12 | dftr2 4089 | . . . 4 | |
13 | 11, 12 | mpbir 145 | . . 3 |
14 | ssrab2 3232 | . . 3 | |
15 | ord0 4376 | . . . . 5 | |
16 | ordsucim 4484 | . . . . 5 | |
17 | 15, 16 | ax-mp 5 | . . . 4 |
18 | suc0 4396 | . . . . 5 | |
19 | ordeq 4357 | . . . . 5 | |
20 | 18, 19 | ax-mp 5 | . . . 4 |
21 | 17, 20 | mpbi 144 | . . 3 |
22 | trssord 4365 | . . 3 | |
23 | 13, 14, 21, 22 | mp3an 1332 | . 2 |
24 | p0ex 4174 | . . . 4 | |
25 | 24 | rabex 4133 | . . 3 |
26 | 25 | elon 4359 | . 2 |
27 | 23, 26 | mpbir 145 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wal 1346 wceq 1348 wcel 2141 crab 2452 wss 3121 c0 3414 csn 3583 wtr 4087 word 4347 con0 4348 csuc 4350 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-uni 3797 df-tr 4088 df-iord 4351 df-on 4353 df-suc 4356 |
This theorem is referenced by: ordtriexmid 4505 ontriexmidim 4506 ordtri2orexmid 4507 ontr2exmid 4509 onsucsssucexmid 4511 ordsoexmid 4546 0elsucexmid 4549 ordpwsucexmid 4554 unfiexmid 6895 exmidonfinlem 7170 |
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