Definition df-lim 6368

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 6420, dflim3 7838, 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 6364	. 2 wff Lim 𝐴
3	1	word 6362	. . 3 wff Ord 𝐴
4		c0 4321	. . . 4 class ∅
5	1, 4	wne 2938	. . 3 wff 𝐴 ≠ ∅
6	1	cuni 4907	. . . 4 class ∪ 𝐴
7	1, 6	wceq 1539	. . 3 wff 𝐴 = ∪ 𝐴
8	3, 5, 7	w3a 1085	. 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  6375  dflim2  6420  limord  6423  limuni  6424  unizlim  6486  limon  7826  dflim3  7838  nnsuc  7875  onfununi  8343  nlim1  8491  nlim2  8492  dfrdg2  35071  ellimits  35186  onsucuni3  36551  omlimcl2  42293  dflim5  42381