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Mirrors > Home > ILE Home > Th. List > ordelsuc | Unicode version |
Description: A set belongs to an ordinal iff its successor is a subset of the ordinal. Exercise 8 of [TakeutiZaring] p. 42 and its converse. (Contributed by NM, 29-Nov-2003.) |
Ref | Expression |
---|---|
ordelsuc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordsucss 4248 | . . 3 | |
2 | 1 | adantl 271 | . 2 |
3 | sucssel 4179 | . . 3 | |
4 | 3 | adantr 270 | . 2 |
5 | 2, 4 | impbid 127 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wcel 1433 wss 2973 word 4117 csuc 4120 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-sn 3404 df-uni 3602 df-tr 3876 df-iord 4121 df-suc 4126 |
This theorem is referenced by: onsucssi 4250 onsucmin 4251 onsucelsucr 4252 onsucsssucr 4253 onsucsssucexmid 4270 ordgt0ge1 6041 nnsucsssuc 6094 |
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