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Mirrors > Home > ILE Home > Th. List > onsucsssucr | Unicode version |
Description: The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4412. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.) |
Ref | Expression |
---|---|
onsucsssucr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordsucim 4386 | . . 3 | |
2 | ordelsuc 4391 | . . 3 | |
3 | 1, 2 | sylan2 284 | . 2 |
4 | ordtr 4270 | . . . 4 | |
5 | trsucss 4315 | . . . 4 | |
6 | 4, 5 | syl 14 | . . 3 |
7 | 6 | adantl 275 | . 2 |
8 | 3, 7 | sylbird 169 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wcel 1465 wss 3041 wtr 3996 word 4254 con0 4255 csuc 4257 |
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-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-sn 3503 df-uni 3707 df-tr 3997 df-iord 4258 df-suc 4263 |
This theorem is referenced by: nnsucsssuc 6356 |
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