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Theorem onsucssi 4523
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 4388 . 2  |-  Ord  B
4 ordelsuc 4502 . 2  |-  ( ( A  e.  On  /\  Ord  B )  ->  ( A  e.  B  <->  suc  A  C_  B ) )
51, 3, 4mp2an 656 1  |-  ( A  e.  B  <->  suc  A  C_  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    e. wcel 1621    C_ wss 3078   Ord word 4284   Oncon0 4285   suc csuc 4287
This theorem is referenced by:  omopthlem1  6539  rankval4  7423  rankc1  7426  rankc2  7427  rankxplim  7433  rankxplim3  7435  onsucsuccmpi  24056
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-tr 4011  df-eprel 4198  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-suc 4291
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