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Theorem ordelsuc 4489
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 4488 . . 3  |-  ( Ord 
B  ->  ( A  e.  B  ->  suc  A  C_  B ) )
21adantl 275 . 2  |-  ( ( A  e.  C  /\  Ord  B )  ->  ( A  e.  B  ->  suc 
A  C_  B )
)
3 sucssel 4409 . . 3  |-  ( A  e.  C  ->  ( suc  A  C_  B  ->  A  e.  B ) )
43adantr 274 . 2  |-  ( ( A  e.  C  /\  Ord  B )  ->  ( suc  A  C_  B  ->  A  e.  B ) )
52, 4impbid 128 1  |-  ( ( A  e.  C  /\  Ord  B )  ->  ( A  e.  B  <->  suc  A  C_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    e. wcel 2141    C_ wss 3121   Ord word 4347   suc csuc 4350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-uni 3797  df-tr 4088  df-iord 4351  df-suc 4356
This theorem is referenced by:  onsucssi  4490  onsucmin  4491  onsucelsucr  4492  onsucsssucr  4493  onsucsssucexmid  4511  frecsuclem  6385  ordgt0ge1  6414  nnsucsssuc  6471  ennnfonelemk  12355  nninfsellemeq  14047
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