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Theorem ordelsuc 4611
Description: A set belongs to an ordinal iff its successor is a subset of the ordinal. Exercise 8 of [TakeutiZaring] p. 42 and its converse. (Contributed by NM, 29-Nov-2003.)
Assertion
Ref Expression
ordelsuc  |-  ( ( A  e.  C  /\  Ord  B )  ->  ( A  e.  B  <->  suc  A  C_  B ) )

Proof of Theorem ordelsuc
StepHypRef Expression
1 ordsucss 4609 . . 3  |-  ( Ord 
B  ->  ( A  e.  B  ->  suc  A  C_  B ) )
21adantl 452 . 2  |-  ( ( A  e.  C  /\  Ord  B )  ->  ( A  e.  B  ->  suc 
A  C_  B )
)
3 sucssel 4485 . . 3  |-  ( A  e.  C  ->  ( suc  A  C_  B  ->  A  e.  B ) )
43adantr 451 . 2  |-  ( ( A  e.  C  /\  Ord  B )  ->  ( suc  A  C_  B  ->  A  e.  B ) )
52, 4impbid 183 1  |-  ( ( A  e.  C  /\  Ord  B )  ->  ( A  e.  B  <->  suc  A  C_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1684    C_ wss 3152   Ord word 4391   suc csuc 4394
This theorem is referenced by:  onsucmin  4612  onsucssi  4632  tfindsg2  4652  ordgt0ge1  6496  onomeneq  7050  omsucdomOLD  7056  cantnflem1  7391  r1ordg  7450  r1val1  7458  rankonidlem  7500  rankxplim3  7551
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 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
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 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398
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