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Theorem onsucelsucr 4280
Description: Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4301. However, the converse does hold where  B is a natural number, as seen at nnsucelsuc 6155. (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 2619 . . . 4  |-  ( suc 
A  e.  suc  B  ->  suc  A  e.  _V )
2 sucexb 4269 . . . 4  |-  ( A  e.  _V  <->  suc  A  e. 
_V )
31, 2sylibr 132 . . 3  |-  ( suc 
A  e.  suc  B  ->  A  e.  _V )
4 onelss 4170 . . . . . . 7  |-  ( B  e.  On  ->  ( suc  A  e.  B  ->  suc  A  C_  B )
)
5 eqimss 3060 . . . . . . . 8  |-  ( suc 
A  =  B  ->  suc  A  C_  B )
65a1i 9 . . . . . . 7  |-  ( B  e.  On  ->  ( suc  A  =  B  ->  suc  A  C_  B )
)
74, 6jaod 670 . . . . . 6  |-  ( B  e.  On  ->  (
( suc  A  e.  B  \/  suc  A  =  B )  ->  suc  A 
C_  B ) )
87adantl 271 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  On )  ->  ( ( suc  A  e.  B  \/  suc  A  =  B )  ->  suc  A  C_  B )
)
9 elsucg 4187 . . . . . . 7  |-  ( suc 
A  e.  _V  ->  ( suc  A  e.  suc  B  <-> 
( suc  A  e.  B  \/  suc  A  =  B ) ) )
102, 9sylbi 119 . . . . . 6  |-  ( A  e.  _V  ->  ( suc  A  e.  suc  B  <->  ( suc  A  e.  B  \/  suc  A  =  B ) ) )
1110adantr 270 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  On )  ->  ( suc  A  e. 
suc  B  <->  ( suc  A  e.  B  \/  suc  A  =  B ) ) )
12 eloni 4158 . . . . . 6  |-  ( B  e.  On  ->  Ord  B )
13 ordelsuc 4277 . . . . . 6  |-  ( ( A  e.  _V  /\  Ord  B )  ->  ( A  e.  B  <->  suc  A  C_  B ) )
1412, 13sylan2 280 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  On )  ->  ( A  e.  B  <->  suc 
A  C_  B )
)
158, 11, 143imtr4d 201 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  On )  ->  ( suc  A  e. 
suc  B  ->  A  e.  B ) )
1615impancom 256 . . 3  |-  ( ( A  e.  _V  /\  suc  A  e.  suc  B
)  ->  ( B  e.  On  ->  A  e.  B ) )
173, 16mpancom 413 . 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 102    <-> wb 103    \/ wo 662    = wceq 1285    e. wcel 1434   _Vcvv 2610    C_ wss 2982   Ord word 4145   Oncon0 4146   suc csuc 4148
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-uni 3622  df-tr 3896  df-iord 4149  df-on 4151  df-suc 4154
This theorem is referenced by:  nnsucelsuc  6155
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