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Theorem onsucmin 4393
Description: The successor of an ordinal number is the smallest larger ordinal number. (Contributed by NM, 28-Nov-2003.)
Assertion
Ref Expression
onsucmin  |-  ( A  e.  On  ->  suc  A  =  |^| { x  e.  On  |  A  e.  x } )
Distinct variable group:    x, A

Proof of Theorem onsucmin
StepHypRef Expression
1 eloni 4267 . . . . 5  |-  ( x  e.  On  ->  Ord  x )
2 ordelsuc 4391 . . . . 5  |-  ( ( A  e.  On  /\  Ord  x )  ->  ( A  e.  x  <->  suc  A  C_  x ) )
31, 2sylan2 284 . . . 4  |-  ( ( A  e.  On  /\  x  e.  On )  ->  ( A  e.  x  <->  suc 
A  C_  x )
)
43rabbidva 2648 . . 3  |-  ( A  e.  On  ->  { x  e.  On  |  A  e.  x }  =  {
x  e.  On  |  suc  A  C_  x }
)
54inteqd 3746 . 2  |-  ( A  e.  On  ->  |^| { x  e.  On  |  A  e.  x }  =  |^| { x  e.  On  |  suc  A  C_  x }
)
6 sucelon 4389 . . 3  |-  ( A  e.  On  <->  suc  A  e.  On )
7 intmin 3761 . . 3  |-  ( suc 
A  e.  On  ->  |^|
{ x  e.  On  |  suc  A  C_  x }  =  suc  A )
86, 7sylbi 120 . 2  |-  ( A  e.  On  ->  |^| { x  e.  On  |  suc  A  C_  x }  =  suc  A )
95, 8eqtr2d 2151 1  |-  ( A  e.  On  ->  suc  A  =  |^| { x  e.  On  |  A  e.  x } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1316    e. wcel 1465   {crab 2397    C_ wss 3041   |^|cint 3741   Ord word 4254   Oncon0 4255   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-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  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-rex 2399  df-rab 2402  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-uni 3707  df-int 3742  df-tr 3997  df-iord 4258  df-on 4260  df-suc 4263
This theorem is referenced by: (None)
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