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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 4437 or weak linearity in ordsoexmid 4477) 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 2837 | . . . . . . . . 9 | |
3 | velsn 3544 | . . . . . . . . 9 | |
4 | 2, 3 | sylib 121 | . . . . . . . 8 |
5 | noel 3367 | . . . . . . . . 9 | |
6 | eleq2 2203 | . . . . . . . . 9 | |
7 | 5, 6 | mtbiri 664 | . . . . . . . 8 |
8 | 4, 7 | syl 14 | . . . . . . 7 |
9 | 8 | adantl 275 | . . . . . 6 |
10 | 1, 9 | pm2.21dd 609 | . . . . 5 |
11 | 10 | gen2 1426 | . . . 4 |
12 | dftr2 4028 | . . . 4 | |
13 | 11, 12 | mpbir 145 | . . 3 |
14 | ssrab2 3182 | . . 3 | |
15 | ord0 4313 | . . . . 5 | |
16 | ordsucim 4416 | . . . . 5 | |
17 | 15, 16 | ax-mp 5 | . . . 4 |
18 | suc0 4333 | . . . . 5 | |
19 | ordeq 4294 | . . . . 5 | |
20 | 18, 19 | ax-mp 5 | . . . 4 |
21 | 17, 20 | mpbi 144 | . . 3 |
22 | trssord 4302 | . . 3 | |
23 | 13, 14, 21, 22 | mp3an 1315 | . 2 |
24 | p0ex 4112 | . . . 4 | |
25 | 24 | rabex 4072 | . . 3 |
26 | 25 | elon 4296 | . 2 |
27 | 23, 26 | mpbir 145 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wal 1329 wceq 1331 wcel 1480 crab 2420 wss 3071 c0 3363 csn 3527 wtr 4026 word 4284 con0 4285 csuc 4287 |
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-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-uni 3737 df-tr 4027 df-iord 4288 df-on 4290 df-suc 4293 |
This theorem is referenced by: ordtriexmid 4437 ordtri2orexmid 4438 ontr2exmid 4440 onsucsssucexmid 4442 ordsoexmid 4477 0elsucexmid 4480 ordpwsucexmid 4485 unfiexmid 6806 exmidonfinlem 7049 |
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