Theorem dflim3OLD 3931

Description: An alternate definition of a limit ordinal, which is any ordinal that is neither zero nor a successor.

Assertion

Ref	Expression
dflim3OLD	|- (Lim A <-> (Ord A /\ -. (A = (/) \/ E.x e. On A = suc x)))

Distinct variable group:   x,A

Proof of Theorem dflim3OLD

Step	Hyp	Ref	Expression
1		limord 3723	. . 3 |- (Lim A -> Ord A)
2		ioran 331	. . . 4 |- (-. (A = (/) \/ E.x e. On A = suc x) <-> (-. A = (/) /\ -. E.x e. On A = suc x))
3		nlim0 3721	. . . . . 6 |- -. Lim (/)
4		limeq 3669	. . . . . 6 |- (A = (/) -> (Lim A <-> Lim (/)))
5	3, 4	mtbiri 785	. . . . 5 |- (A = (/) -> -. Lim A)
6	5	con2i 113	. . . 4 |- (Lim A -> -. A = (/))
7		limuni 3724	. . . . 5 |- (Lim A -> A = U.A)
8		orduninsuc 3925	. . . . . 6 |- (Ord A -> (A = U.A <-> -. E.x e. On A = suc x))
9	1, 8	syl 12	. . . . 5 |- (Lim A -> (A = U.A <-> -. E.x e. On A = suc x))
10	7, 9	mpbid 212	. . . 4 |- (Lim A -> -. E.x e. On A = suc x)
11	2, 6, 10	sylanbrc 527	. . 3 |- (Lim A -> -. (A = (/) \/ E.x e. On A = suc x))
12	1, 11	jca 310	. 2 |- (Lim A -> (Ord A /\ -. (A = (/) \/ E.x e. On A = suc x)))
13		ordzsl 3927	. . . . 5 |- (Ord A <-> (A = (/) \/ E.x e. On A = suc x \/ Lim A))
14	13	biimpi 168	. . . 4 |- (Ord A -> (A = (/) \/ E.x e. On A = suc x \/ Lim A))
15		df-3or 859	. . . 4 |- ((A = (/) \/ E.x e. On A = suc x \/ Lim A) <-> ((A = (/) \/ E.x e. On A = suc x) \/ Lim A))
16	14, 15	sylib 215	. . 3 |- (Ord A -> ((A = (/) \/ E.x e. On A = suc x) \/ Lim A))
17	16	orcanai 754	. 2 |- ((Ord A /\ -. (A = (/) \/ E.x e. On A = suc x)) -> Lim A)
18	12, 17	impbii 174	1 |- (Lim A <-> (Ord A /\ -. (A = (/) \/ E.x e. On A = suc x)))

Syntax hints:  -. wn 2   <-> wb 163   \/ wo 239   /\ wa 240   \/ w3o 857   = wceq 1298  E.wrex 2106  (/)c0 2875  U.cuni 3177  Ord word 3656  Oncon0 3657  Lim wlim 3658  suc csuc 3659

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663

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