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Theorem onsucssi 4631
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 4496 . 2  |-  Ord  B
4 ordelsuc 4610 . 2  |-  ( ( A  e.  On  /\  Ord  B )  ->  ( A  e.  B  <->  suc  A  C_  B ) )
51, 3, 4mp2an 653 1  |-  ( A  e.  B  <->  suc  A  C_  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    e. wcel 1685    C_ wss 3153   Ord word 4390   Oncon0 4391   suc csuc 4393
This theorem is referenced by:  omopthlem1  6649  rankval4  7535  rankc1  7538  rankc2  7539  rankxplim  7545  rankxplim3  7547  onsucsuccmpi  24292
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-tr 4115  df-eprel 4304  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-suc 4397
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