Theorem dflim3 3113

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 3023	. . 3 |- (Lim A -> Ord A)
2		nlim0 3022	. . . . . . 7 |- -. Lim (/)
3		limeq 2955	. . . . . . 7 |- (A = (/) -> (Lim A <-> Lim (/)))
4	2, 3	mtbiri 716	. . . . . 6 |- (A = (/) -> -. Lim A)
5	4	con2i 97	. . . . 5 |- (Lim A -> -. A = (/))
6		limuni 3024	. . . . . 6 |- (Lim A -> A = U.A)
7		orduninsuc 3109	. . . . . . 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 3111	. . . . 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 775	. . . 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 689	. 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 773   = wceq 954  E.wrex 1643  (/)c0 2276  U.cuni 2498  Ord word 2942  Oncon0 2943  Lim wlim 2944  suc csuc 2945

This theorem is referenced by:  nlimon 3117  oalimcl 4184  omlimcl 4199

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949

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