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Theorem onsucelsucr 4432
Description: Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4453. However, the converse does hold where  B is a natural number, as seen at nnsucelsuc 6395. (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 2700 . . . 4  |-  ( suc 
A  e.  suc  B  ->  suc  A  e.  _V )
2 sucexb 4421 . . . 4  |-  ( A  e.  _V  <->  suc  A  e. 
_V )
31, 2sylibr 133 . . 3  |-  ( suc 
A  e.  suc  B  ->  A  e.  _V )
4 onelss 4317 . . . . . . 7  |-  ( B  e.  On  ->  ( suc  A  e.  B  ->  suc  A  C_  B )
)
5 eqimss 3156 . . . . . . . 8  |-  ( suc 
A  =  B  ->  suc  A  C_  B )
65a1i 9 . . . . . . 7  |-  ( B  e.  On  ->  ( suc  A  =  B  ->  suc  A  C_  B )
)
74, 6jaod 707 . . . . . 6  |-  ( B  e.  On  ->  (
( suc  A  e.  B  \/  suc  A  =  B )  ->  suc  A 
C_  B ) )
87adantl 275 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  On )  ->  ( ( suc  A  e.  B  \/  suc  A  =  B )  ->  suc  A  C_  B )
)
9 elsucg 4334 . . . . . . 7  |-  ( suc 
A  e.  _V  ->  ( suc  A  e.  suc  B  <-> 
( suc  A  e.  B  \/  suc  A  =  B ) ) )
102, 9sylbi 120 . . . . . 6  |-  ( A  e.  _V  ->  ( suc  A  e.  suc  B  <->  ( suc  A  e.  B  \/  suc  A  =  B ) ) )
1110adantr 274 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  On )  ->  ( suc  A  e. 
suc  B  <->  ( suc  A  e.  B  \/  suc  A  =  B ) ) )
12 eloni 4305 . . . . . 6  |-  ( B  e.  On  ->  Ord  B )
13 ordelsuc 4429 . . . . . 6  |-  ( ( A  e.  _V  /\  Ord  B )  ->  ( A  e.  B  <->  suc  A  C_  B ) )
1412, 13sylan2 284 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  On )  ->  ( A  e.  B  <->  suc 
A  C_  B )
)
158, 11, 143imtr4d 202 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  On )  ->  ( suc  A  e. 
suc  B  ->  A  e.  B ) )
1615impancom 258 . . 3  |-  ( ( A  e.  _V  /\  suc  A  e.  suc  B
)  ->  ( B  e.  On  ->  A  e.  B ) )
173, 16mpancom 419 . 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 103    <-> wb 104    \/ wo 698    = wceq 1332    e. wcel 1481   _Vcvv 2689    C_ wss 3076   Ord word 4292   Oncon0 4293   suc csuc 4295
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-uni 3745  df-tr 4035  df-iord 4296  df-on 4298  df-suc 4301
This theorem is referenced by:  nnsucelsuc  6395
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