Definition df-lim 6253

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 6304, dflim3 7666, 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 6249	. 2 wff Lim 𝐴
3	1	word 6247	. . 3 wff Ord 𝐴
4		c0 4254	. . . 4 class ∅
5	1, 4	wne 2943	. . 3 wff 𝐴 ≠ ∅
6	1	cuni 4836	. . . 4 class ∪ 𝐴
7	1, 6	wceq 1543	. . 3 wff 𝐴 = ∪ 𝐴
8	3, 5, 7	w3a 1089	. 2 wff (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴)
9	2, 8	wb 209	1 wff (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴))

This definition is referenced by:  limeq  6260  dflim2  6304  limord  6307  limuni  6308  unizlim  6365  limon  7655  dflim3  7666  nnsuc  7702  onfununi  8120  dfrdg2  33652  ellimits  34114  onsucuni3  35444