Theorem dflim3 3108

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

Assertion

Ref	Expression
dflim3	|- (Lim A <-> (Ord A /\ -. (A = (/) \/ E.x e. On A = suc x)))

Distinct variable group:   x,A

Proof of Theorem dflim3

Step	Hyp	Ref	Expression
1		limord 3018	. . 3 |- (Lim A -> Ord A)
2		nlim0 3017	. . . . . . 7 |- -. Lim (/)
3		limeq 2950	. . . . . . 7 |- (A = (/) -> (Lim A <-> Lim (/)))
4	2, 3	mtbiri 715	. . . . . 6 |- (A = (/) -> -. Lim A)
5	4	con2i 97	. . . . 5 |- (Lim A -> -. A = (/))
6		limuni 3019	. . . . . 6 |- (Lim A -> A = U.A)
7		orduninsuc 3104	. . . . . . 7 |- (Ord A -> (A = U.A <-> -. E.x e. On A = suc x))
8	1, 7	syl 10	. . . . . 6 |- (Lim A -> (A = U.A <-> -. E.x e. On A = suc x))
9	6, 8	mpbid 195	. . . . 5 |- (Lim A -> -. E.x e. On A = suc x)
10	5, 9	jca 288	. . . 4 |- (Lim A -> (-. A = (/) /\ -. E.x e. On A = suc x))
11		ioran 306	. . . 4 |- (-. (A = (/) \/ E.x e. On A = suc x) <-> (-. A = (/) /\ -. E.x e. On A = suc x))
12	10, 11	sylibr 200	. . 3 |- (Lim A -> -. (A = (/) \/ E.x e. On A = suc x))
13	1, 12	jca 288	. 2 |- (Lim A -> (Ord A /\ -. (A = (/) \/ E.x e. On A = suc x)))
14		ordzsl 3106	. . . . 5 |- (Ord A <-> (A = (/) \/ E.x e. On A = suc x \/ Lim A))
15	14	biimp 151	. . . 4 |- (Ord A -> (A = (/) \/ E.x e. On A = suc x \/ Lim A))
16		df-3or 774	. . . 4 |- ((A = (/) \/ E.x e. On A = suc x \/ Lim A) <-> ((A = (/) \/ E.x e. On A = suc x) \/ Lim A))
17	15, 16	sylib 198	. . 3 |- (Ord A -> ((A = (/) \/ E.x e. On A = suc x) \/ Lim A))
18	17	orcanai 688	. 2 |- ((Ord A /\ -. (A = (/) \/ E.x e. On A = suc x)) -> Lim A)
19	13, 18	impbi 157	1 |- (Lim A <-> (Ord A /\ -. (A = (/) \/ E.x e. On A = suc x)))

Syntax hints:  -. wn 2   <-> wb 146   \/ wo 222   /\ wa 223   \/ w3o 772   = wceq 953  E.wrex 1638  (/)c0 2270  U.cuni 2493  Ord word 2937  Oncon0 2938  Lim wlim 2939  suc csuc 2940

This theorem is referenced by:  nlimon 3112  oalimcl 4178  omlimcl 4193

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944

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