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Definition df-lim 5645
 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 5698, dflim3 6939, 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 5641 . 2 wff Lim 𝐴
31word 5639 . . 3 wff Ord 𝐴
4 c0 3874 . . . 4 class
51, 4wne 2780 . . 3 wff 𝐴 ≠ ∅
61cuni 4372 . . . 4 class 𝐴
71, 6wceq 1475 . . 3 wff 𝐴 = 𝐴
83, 5, 7w3a 1031 . 2 wff (Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴)
92, 8wb 195 1 wff (Lim 𝐴 ↔ (Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴))
 Colors of variables: wff setvar class This definition is referenced by:  limeq  5652  dflim2  5698  limord  5701  limuni  5702  unizlim  5761  limon  6928  dflim3  6939  nnsuc  6974  onfununi  7325  dfrdg2  30945  ellimits  31187  onsucuni3  32391
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