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Theorem onsucmin 4502
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 4371 . . . . 5  |-  ( x  e.  On  ->  Ord  x )
2 ordelsuc 4500 . . . . 5  |-  ( ( A  e.  On  /\  Ord  x )  ->  ( A  e.  x  <->  suc  A  C_  x ) )
31, 2sylan2 286 . . . 4  |-  ( ( A  e.  On  /\  x  e.  On )  ->  ( A  e.  x  <->  suc 
A  C_  x )
)
43rabbidva 2725 . . 3  |-  ( A  e.  On  ->  { x  e.  On  |  A  e.  x }  =  {
x  e.  On  |  suc  A  C_  x }
)
54inteqd 3847 . 2  |-  ( A  e.  On  ->  |^| { x  e.  On  |  A  e.  x }  =  |^| { x  e.  On  |  suc  A  C_  x }
)
6 onsucb 4498 . . 3  |-  ( A  e.  On  <->  suc  A  e.  On )
7 intmin 3862 . . 3  |-  ( suc 
A  e.  On  ->  |^|
{ x  e.  On  |  suc  A  C_  x }  =  suc  A )
86, 7sylbi 121 . 2  |-  ( A  e.  On  ->  |^| { x  e.  On  |  suc  A  C_  x }  =  suc  A )
95, 8eqtr2d 2211 1  |-  ( A  e.  On  ->  suc  A  =  |^| { x  e.  On  |  A  e.  x } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1353    e. wcel 2148   {crab 2459    C_ wss 3129   |^|cint 3842   Ord word 4358   Oncon0 4359   suc csuc 4361
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-uni 3808  df-int 3843  df-tr 4099  df-iord 4362  df-on 4364  df-suc 4367
This theorem is referenced by: (None)
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