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Theorem ordelsuc 4391
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 4390 . . 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 4316 . . 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 1465    C_ wss 3041   Ord word 4254   suc csuc 4257
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-uni 3707  df-tr 3997  df-iord 4258  df-suc 4263
This theorem is referenced by:  onsucssi  4392  onsucmin  4393  onsucelsucr  4394  onsucsssucr  4395  onsucsssucexmid  4412  frecsuclem  6271  ordgt0ge1  6300  nnsucsssuc  6356  ennnfonelemk  11840  nninfsellemeq  13137
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