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Theorem onsucssi 4754
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
onssi.1  |-  A  e.  On
onsucssi.2  |-  B  e.  On
Assertion
Ref Expression
onsucssi  |-  ( A  e.  B  <->  suc  A  C_  B )

Proof of Theorem onsucssi
StepHypRef Expression
1 onssi.1 . 2  |-  A  e.  On
2 onsucssi.2 . . 3  |-  B  e.  On
32onordi 4619 . 2  |-  Ord  B
4 ordelsuc 4733 . 2  |-  ( ( A  e.  On  /\  Ord  B )  ->  ( A  e.  B  <->  suc  A  C_  B ) )
51, 3, 4mp2an 654 1  |-  ( A  e.  B  <->  suc  A  C_  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    e. wcel 1717    C_ wss 3256   Ord word 4514   Oncon0 4515   suc csuc 4517
This theorem is referenced by:  omopthlem1  6827  rankval4  7719  rankc1  7722  rankc2  7723  rankxplim  7729  rankxplim3  7731  onsucsuccmpi  25900
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-tr 4237  df-eprel 4428  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-suc 4521
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