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Definition df-lim 4413
Description: Define the limit ordinal predicate, which is true for a non-empty ordinal that is not a successor (i.e. that is the union of itself). Our definition combines the definition of Lim of [BellMachover] p. 471 and Exercise 1 of [TakeutiZaring] p. 42. See dflim2 4464, dflim3 4654, and dflim4 for alternate definitions. (Contributed by NM, 22-Apr-1994.)
Assertion
Ref Expression
df-lim  |-  ( Lim 
A  <->  ( Ord  A  /\  A  =/=  (/)  /\  A  =  U. A ) )

Detailed syntax breakdown of Definition df-lim
StepHypRef Expression
1 cA . . 3  class  A
21wlim 4409 . 2  wff  Lim  A
31word 4407 . . 3  wff  Ord  A
4 c0 3468 . . . 4  class  (/)
51, 4wne 2459 . . 3  wff  A  =/=  (/)
61cuni 3843 . . . 4  class  U. A
71, 6wceq 1632 . . 3  wff  A  = 
U. A
83, 5, 7w3a 934 . 2  wff  ( Ord 
A  /\  A  =/=  (/) 
/\  A  =  U. A )
92, 8wb 176 1  wff  ( Lim 
A  <->  ( Ord  A  /\  A  =/=  (/)  /\  A  =  U. A ) )
Colors of variables: wff set class
This definition is referenced by:  limeq  4420  dflim2  4464  limord  4467  limuni  4468  unizlim  4525  limon  4643  dflim3  4654  nnsuc  4689  onfununi  6374  abianfplem  6486  dfrdg2  24223  ellimits  24521
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