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Mirrors > Home > ILE Home > Th. List > ordtriexmidlem2 | 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 helps connect that set to excluded middle. (Contributed by Jim Kingdon, 28-Jan-2019.) |
Ref | Expression |
---|---|
ordtriexmidlem2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 3367 | . . 3 | |
2 | eleq2 2203 | . . 3 | |
3 | 1, 2 | mtbiri 664 | . 2 |
4 | 0ex 4055 | . . . 4 | |
5 | 4 | snid 3556 | . . 3 |
6 | biidd 171 | . . . 4 | |
7 | 6 | elrab3 2841 | . . 3 |
8 | 5, 7 | ax-mp 5 | . 2 |
9 | 3, 8 | sylnib 665 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wb 104 wceq 1331 wcel 1480 crab 2420 c0 3363 csn 3527 |
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 2121 ax-nul 4054 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rab 2425 df-v 2688 df-dif 3073 df-nul 3364 df-sn 3533 |
This theorem is referenced by: ordtriexmid 4437 ordtri2orexmid 4438 ontr2exmid 4440 onsucsssucexmid 4442 ordsoexmid 4477 0elsucexmid 4480 ordpwsucexmid 4485 |
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