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Theorem onsucmin 6641
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 4878 . . . . 5  |-  ( x  e.  On  ->  Ord  x )
2 ordelsuc 6640 . . . . 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 3086 . . 3  |-  ( A  e.  On  ->  { x  e.  On  |  A  e.  x }  =  {
x  e.  On  |  suc  A  C_  x }
)
54inteqd 4276 . 2  |-  ( A  e.  On  ->  |^| { x  e.  On  |  A  e.  x }  =  |^| { x  e.  On  |  suc  A  C_  x }
)
6 sucelon 6637 . . 3  |-  ( A  e.  On  <->  suc  A  e.  On )
7 intmin 4291 . . 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 2485 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 1383    e. wcel 1804   {crab 2797    C_ wss 3461   |^|cint 4271   Ord word 4867   Oncon0 4868   suc csuc 4870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-br 4438  df-opab 4496  df-tr 4531  df-eprel 4781  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-suc 4874
This theorem is referenced by:  ranksnb  8248
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