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Theorem onsucelsucr 4525
Description: Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4547. However, the converse does hold where  B is a natural number, as seen at nnsucelsuc 6517. (Contributed by Jim Kingdon, 17-Jul-2019.)
Assertion
Ref Expression
onsucelsucr  |-  ( B  e.  On  ->  ( suc  A  e.  suc  B  ->  A  e.  B ) )

Proof of Theorem onsucelsucr
StepHypRef Expression
1 elex 2763 . . . 4  |-  ( suc 
A  e.  suc  B  ->  suc  A  e.  _V )
2 sucexb 4514 . . . 4  |-  ( A  e.  _V  <->  suc  A  e. 
_V )
31, 2sylibr 134 . . 3  |-  ( suc 
A  e.  suc  B  ->  A  e.  _V )
4 onelss 4405 . . . . . . 7  |-  ( B  e.  On  ->  ( suc  A  e.  B  ->  suc  A  C_  B )
)
5 eqimss 3224 . . . . . . . 8  |-  ( suc 
A  =  B  ->  suc  A  C_  B )
65a1i 9 . . . . . . 7  |-  ( B  e.  On  ->  ( suc  A  =  B  ->  suc  A  C_  B )
)
74, 6jaod 718 . . . . . 6  |-  ( B  e.  On  ->  (
( suc  A  e.  B  \/  suc  A  =  B )  ->  suc  A 
C_  B ) )
87adantl 277 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  On )  ->  ( ( suc  A  e.  B  \/  suc  A  =  B )  ->  suc  A  C_  B )
)
9 elsucg 4422 . . . . . . 7  |-  ( suc 
A  e.  _V  ->  ( suc  A  e.  suc  B  <-> 
( suc  A  e.  B  \/  suc  A  =  B ) ) )
102, 9sylbi 121 . . . . . 6  |-  ( A  e.  _V  ->  ( suc  A  e.  suc  B  <->  ( suc  A  e.  B  \/  suc  A  =  B ) ) )
1110adantr 276 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  On )  ->  ( suc  A  e. 
suc  B  <->  ( suc  A  e.  B  \/  suc  A  =  B ) ) )
12 eloni 4393 . . . . . 6  |-  ( B  e.  On  ->  Ord  B )
13 ordelsuc 4522 . . . . . 6  |-  ( ( A  e.  _V  /\  Ord  B )  ->  ( A  e.  B  <->  suc  A  C_  B ) )
1412, 13sylan2 286 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  On )  ->  ( A  e.  B  <->  suc 
A  C_  B )
)
158, 11, 143imtr4d 203 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  On )  ->  ( suc  A  e. 
suc  B  ->  A  e.  B ) )
1615impancom 260 . . 3  |-  ( ( A  e.  _V  /\  suc  A  e.  suc  B
)  ->  ( B  e.  On  ->  A  e.  B ) )
173, 16mpancom 422 . 2  |-  ( suc 
A  e.  suc  B  ->  ( B  e.  On  ->  A  e.  B ) )
1817com12 30 1  |-  ( B  e.  On  ->  ( suc  A  e.  suc  B  ->  A  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 709    = wceq 1364    e. wcel 2160   _Vcvv 2752    C_ wss 3144   Ord word 4380   Oncon0 4381   suc csuc 4383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-uni 3825  df-tr 4117  df-iord 4384  df-on 4386  df-suc 4389
This theorem is referenced by:  nnsucelsuc  6517
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