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Definition df-lim 5948
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 6000, dflim3 7280, 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 5944 . 2 wff Lim 𝐴
31word 5942 . . 3 wff Ord 𝐴
4 c0 4123 . . . 4 class
51, 4wne 2985 . . 3 wff 𝐴 ≠ ∅
61cuni 4637 . . . 4 class 𝐴
71, 6wceq 1637 . . 3 wff 𝐴 = 𝐴
83, 5, 7w3a 1100 . 2 wff (Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴)
92, 8wb 197 1 wff (Lim 𝐴 ↔ (Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴))
Colors of variables: wff setvar class
This definition is referenced by:  limeq  5955  dflim2  6000  limord  6003  limuni  6004  unizlim  6060  limon  7269  dflim3  7280  nnsuc  7315  onfununi  7677  dfrdg2  32026  ellimits  32343  onsucuni3  33533
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