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Theorem onsucssi 4422
 Description: A set belongs to an ordinal number iff its successor is a subset of the ordinal number. Exercise 8 of [TakeutiZaring] p. 42 and its converse. (Contributed by NM, 16-Sep-1995.)
Hypotheses
Ref Expression
onsucssi.1
onsucssi.2
Assertion
Ref Expression
onsucssi

Proof of Theorem onsucssi
StepHypRef Expression
1 onsucssi.1 . 2
2 onsucssi.2 . . 3
32onordi 4348 . 2
4 ordelsuc 4421 . 2
51, 3, 4mp2an 422 1
 Colors of variables: wff set class Syntax hints:   wb 104   wcel 1480   wss 3071   word 4284  con0 4285   csuc 4287 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 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 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-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-uni 3737  df-tr 4027  df-iord 4288  df-on 4290  df-suc 4293 This theorem is referenced by: (None)
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