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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 4486 | . . 3 | |
2 | 1 | adantl 275 | . 2 |
3 | sucssel 4407 | . . 3 | |
4 | 3 | adantr 274 | . 2 |
5 | 2, 4 | impbid 128 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wcel 2141 wss 3121 word 4345 csuc 4348 |
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 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-ext 2152 |
This theorem depends on definitions: df-bi 116 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-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-sn 3587 df-uni 3795 df-tr 4086 df-iord 4349 df-suc 4354 |
This theorem is referenced by: onsucssi 4488 onsucmin 4489 onsucelsucr 4490 onsucsssucr 4491 onsucsssucexmid 4509 frecsuclem 6382 ordgt0ge1 6411 nnsucsssuc 6468 ennnfonelemk 12342 nninfsellemeq 13969 |
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