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Theorem ordelsuc 4569
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 4567 . . 3  |-  ( Ord 
B  ->  ( A  e.  B  ->  suc  A  C_  B ) )
21adantl 454 . 2  |-  ( ( A  e.  C  /\  Ord  B )  ->  ( A  e.  B  ->  suc 
A  C_  B )
)
3 sucssel 4443 . . 3  |-  ( A  e.  C  ->  ( suc  A  C_  B  ->  A  e.  B ) )
43adantr 453 . 2  |-  ( ( A  e.  C  /\  Ord  B )  ->  ( suc  A  C_  B  ->  A  e.  B ) )
52, 4impbid 185 1  |-  ( ( A  e.  C  /\  Ord  B )  ->  ( A  e.  B  <->  suc  A  C_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    e. wcel 1621    C_ wss 3113   Ord word 4349   suc csuc 4352
This theorem is referenced by:  onsucmin  4570  onsucssi  4590  tfindsg2  4610  ordgt0ge1  6450  onomeneq  7004  omsucdomOLD  7010  cantnflem1  7345  r1ordg  7404  r1val1  7412  rankonidlem  7454  rankxplim3  7505
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 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pr 4172  ax-un 4470
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-rab 2525  df-v 2759  df-sbc 2953  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-br 3984  df-opab 4038  df-tr 4074  df-eprel 4263  df-po 4272  df-so 4273  df-fr 4310  df-we 4312  df-ord 4353  df-on 4354  df-suc 4356
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