Theorem dflim3 3118

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 3028	. . 3 |- (Lim A -> Ord A)
2		nlim0 3027	. . . . . . 7 |- -. Lim (/)
3		limeq 2960	. . . . . . 7 |- (A = (/) -> (Lim A <-> Lim (/)))
4	2, 3	mtbiri 717	. . . . . 6 |- (A = (/) -> -. Lim A)
5	4	con2i 97	. . . . 5 |- (Lim A -> -. A = (/))
6		limuni 3029	. . . . . 6 |- (Lim A -> A = U.A)
7		orduninsuc 3114	. . . . . . 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 3116	. . . . 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 776	. . . 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 690	. 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 774   = wceq 956  E.wrex 1646  (/)c0 2280  U.cuni 2503  Ord word 2947  Oncon0 2948  Lim wlim 2949  suc csuc 2950

This theorem is referenced by:  nlimon 3122  oalimcl 4194  omlimcl 4209

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954

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