Definition df-lim 6366

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 6418, dflim3 7832, and dflim4 for alternate definitions. (Contributed by NM, 22-Apr-1994.)

Assertion

Ref	Expression
df-lim	⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴))

Detailed syntax breakdown of Definition df-lim

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2	1	wlim 6362	. 2 wff Lim 𝐴
3	1	word 6360	. . 3 wff Ord 𝐴
4		c0 4321	. . . 4 class ∅
5	1, 4	wne 2940	. . 3 wff 𝐴 ≠ ∅
6	1	cuni 4907	. . . 4 class ∪ 𝐴
7	1, 6	wceq 1541	. . 3 wff 𝐴 = ∪ 𝐴
8	3, 5, 7	w3a 1087	. 2 wff (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴)
9	2, 8	wb 205	1 wff (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴))

This definition is referenced by:  limeq  6373  dflim2  6418  limord  6421  limuni  6422  unizlim  6484  limon  7820  dflim3  7832  nnsuc  7869  onfununi  8337  nlim1  8485  nlim2  8486  dfrdg2  34755  ellimits  34870  onsucuni3  36236  omlimcl2  41976  dflim5  42064