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Theorem onsucelsucr 4325
Description: Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4346. However, the converse does hold where  B is a natural number, as seen at nnsucelsuc 6252. (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 2630 . . . 4  |-  ( suc 
A  e.  suc  B  ->  suc  A  e.  _V )
2 sucexb 4314 . . . 4  |-  ( A  e.  _V  <->  suc  A  e. 
_V )
31, 2sylibr 132 . . 3  |-  ( suc 
A  e.  suc  B  ->  A  e.  _V )
4 onelss 4214 . . . . . . 7  |-  ( B  e.  On  ->  ( suc  A  e.  B  ->  suc  A  C_  B )
)
5 eqimss 3078 . . . . . . . 8  |-  ( suc 
A  =  B  ->  suc  A  C_  B )
65a1i 9 . . . . . . 7  |-  ( B  e.  On  ->  ( suc  A  =  B  ->  suc  A  C_  B )
)
74, 6jaod 672 . . . . . 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 4231 . . . . . . 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 4202 . . . . . 6  |-  ( B  e.  On  ->  Ord  B )
13 ordelsuc 4322 . . . . . 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 664    = wceq 1289    e. wcel 1438   _Vcvv 2619    C_ wss 2999   Ord word 4189   Oncon0 4190   suc csuc 4192
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-uni 3654  df-tr 3937  df-iord 4193  df-on 4195  df-suc 4198
This theorem is referenced by:  nnsucelsuc  6252
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