Definition df-lim 5697

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 5750, dflim3 7009, 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 5693	. 2 wff Lim 𝐴
3	1	word 5691	. . 3 wff Ord 𝐴
4		c0 3897	. . . 4 class ∅
5	1, 4	wne 2790	. . 3 wff 𝐴 ≠ ∅
6	1	cuni 4409	. . . 4 class ∪ 𝐴
7	1, 6	wceq 1480	. . 3 wff 𝐴 = ∪ 𝐴
8	3, 5, 7	w3a 1036	. 2 wff (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴)
9	2, 8	wb 196	1 wff (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴))

This definition is referenced by:  limeq  5704  dflim2  5750  limord  5753  limuni  5754  unizlim  5813  limon  6998  dflim3  7009  nnsuc  7044  onfununi  7398  dfrdg2  31455  ellimits  31712  onsucuni3  32886