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Theorem onsucmin 6534
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 4829 . . . . 5  |-  ( x  e.  On  ->  Ord  x )
2 ordelsuc 6533 . . . . 5  |-  ( ( A  e.  On  /\  Ord  x )  ->  ( A  e.  x  <->  suc  A  C_  x ) )
31, 2sylan2 474 . . . 4  |-  ( ( A  e.  On  /\  x  e.  On )  ->  ( A  e.  x  <->  suc 
A  C_  x )
)
43rabbidva 3061 . . 3  |-  ( A  e.  On  ->  { x  e.  On  |  A  e.  x }  =  {
x  e.  On  |  suc  A  C_  x }
)
54inteqd 4233 . 2  |-  ( A  e.  On  ->  |^| { x  e.  On  |  A  e.  x }  =  |^| { x  e.  On  |  suc  A  C_  x }
)
6 sucelon 6530 . . 3  |-  ( A  e.  On  <->  suc  A  e.  On )
7 intmin 4248 . . 3  |-  ( suc 
A  e.  On  ->  |^|
{ x  e.  On  |  suc  A  C_  x }  =  suc  A )
86, 7sylbi 195 . 2  |-  ( A  e.  On  ->  |^| { x  e.  On  |  suc  A  C_  x }  =  suc  A )
95, 8eqtr2d 2493 1  |-  ( A  e.  On  ->  suc  A  =  |^| { x  e.  On  |  A  e.  x } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1370    e. wcel 1758   {crab 2799    C_ wss 3428   |^|cint 4228   Ord word 4818   Oncon0 4819   suc csuc 4821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-br 4393  df-opab 4451  df-tr 4486  df-eprel 4732  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-suc 4825
This theorem is referenced by:  ranksnb  8137
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