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Mirrors > Home > MPE Home > Th. List > ordelsuc | Structured version Visualization version 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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordsucss 6671 |
. . 3
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2 | 1 | adantl 472 |
. 2
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3 | sucssel 5533 |
. . 3
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4 | 3 | adantr 471 |
. 2
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5 | 2, 4 | impbid 195 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1679 ax-4 1692 ax-5 1768 ax-6 1815 ax-7 1861 ax-8 1899 ax-9 1906 ax-10 1925 ax-11 1930 ax-12 1943 ax-13 2101 ax-ext 2441 ax-sep 4538 ax-nul 4547 ax-pr 4652 ax-un 6609 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3or 992 df-3an 993 df-tru 1457 df-ex 1674 df-nf 1678 df-sb 1808 df-eu 2313 df-mo 2314 df-clab 2448 df-cleq 2454 df-clel 2457 df-nfc 2591 df-ne 2634 df-ral 2753 df-rex 2754 df-rab 2757 df-v 3058 df-sbc 3279 df-dif 3418 df-un 3420 df-in 3422 df-ss 3429 df-pss 3431 df-nul 3743 df-if 3893 df-sn 3980 df-pr 3982 df-tp 3984 df-op 3986 df-uni 4212 df-br 4416 df-opab 4475 df-tr 4511 df-eprel 4763 df-po 4773 df-so 4774 df-fr 4811 df-we 4813 df-ord 5444 df-on 5445 df-suc 5447 |
This theorem is referenced by: onsucmin 6674 onsucssi 6694 tfindsg2 6714 ordgt0ge1 7224 onomeneq 7787 cantnflem1 8219 r1ordg 8274 r1val1 8282 rankonidlem 8324 rankxplim3 8377 |
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