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Theorem onsucssi 4634
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 4499 . 2  |-  Ord  B
4 ordelsuc 4613 . 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 1686    C_ wss 3154   Ord word 4393   Oncon0 4394   suc csuc 4396
This theorem is referenced by:  omopthlem1  6655  rankval4  7541  rankc1  7544  rankc2  7545  rankxplim  7551  rankxplim3  7553  onsucsuccmpi  24884
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216  ax-un 4514
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-tr 4116  df-eprel 4307  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-suc 4400
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