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Theorem onsucssi 4788
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 4653 . 2  |-  Ord  B
4 ordelsuc 4767 . 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 1721    C_ wss 3288   Ord word 4548   Oncon0 4549   suc csuc 4551
This theorem is referenced by:  omopthlem1  6865  rankval4  7757  rankc1  7760  rankc2  7761  rankxplim  7767  rankxplim3  7769  onsucsuccmpi  26105
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371  ax-un 4668
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-tr 4271  df-eprel 4462  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-suc 4555
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