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Definition df-lim 5766
Description: Define the limit ordinal predicate, which is true for a nonempty 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 5819, dflim3 7089, and dflim4 for alternate definitions. (Contributed by NM, 22-Apr-1994.)
Assertion
Ref Expression
df-lim (Lim 𝐴 ↔ (Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴))

Detailed syntax breakdown of Definition df-lim
StepHypRef Expression
1 cA . . 3 class 𝐴
21wlim 5762 . 2 wff Lim 𝐴
31word 5760 . . 3 wff Ord 𝐴
4 c0 3948 . . . 4 class
51, 4wne 2823 . . 3 wff 𝐴 ≠ ∅
61cuni 4468 . . . 4 class 𝐴
71, 6wceq 1523 . . 3 wff 𝐴 = 𝐴
83, 5, 7w3a 1054 . 2 wff (Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴)
92, 8wb 196 1 wff (Lim 𝐴 ↔ (Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴))
Colors of variables: wff setvar class
This definition is referenced by:  limeq  5773  dflim2  5819  limord  5822  limuni  5823  unizlim  5882  limon  7078  dflim3  7089  nnsuc  7124  onfununi  7483  dfrdg2  31825  ellimits  32142  onsucuni3  33345
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