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Theorem onsucmin 4803
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 4593 . . . . 5  |-  ( x  e.  On  ->  Ord  x )
2 ordelsuc 4802 . . . . 5  |-  ( ( A  e.  On  /\  Ord  x )  ->  ( A  e.  x  <->  suc  A  C_  x ) )
31, 2sylan2 462 . . . 4  |-  ( ( A  e.  On  /\  x  e.  On )  ->  ( A  e.  x  <->  suc 
A  C_  x )
)
43rabbidva 2949 . . 3  |-  ( A  e.  On  ->  { x  e.  On  |  A  e.  x }  =  {
x  e.  On  |  suc  A  C_  x }
)
54inteqd 4057 . 2  |-  ( A  e.  On  ->  |^| { x  e.  On  |  A  e.  x }  =  |^| { x  e.  On  |  suc  A  C_  x }
)
6 sucelon 4799 . . 3  |-  ( A  e.  On  <->  suc  A  e.  On )
7 intmin 4072 . . 3  |-  ( suc 
A  e.  On  ->  |^|
{ x  e.  On  |  suc  A  C_  x }  =  suc  A )
86, 7sylbi 189 . 2  |-  ( A  e.  On  ->  |^| { x  e.  On  |  suc  A  C_  x }  =  suc  A )
95, 8eqtr2d 2471 1  |-  ( A  e.  On  ->  suc  A  =  |^| { x  e.  On  |  A  e.  x } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726   {crab 2711    C_ wss 3322   |^|cint 4052   Ord word 4582   Oncon0 4583   suc csuc 4585
This theorem is referenced by:  ranksnb  7755
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-br 4215  df-opab 4269  df-tr 4305  df-eprel 4496  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-suc 4589
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