Theorem dflim3 3930

Description: An alternate definition of a limit ordinal, which is any ordinal that is neither zero nor a successor. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)

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		df-lim 3662	. 2 |- (Lim A <-> (Ord A /\ A =/= (/) /\ A = U.A))
2		3anass 862	. 2 |- ((Ord A /\ A =/= (/) /\ A = U.A) <-> (Ord A /\ (A =/= (/) /\ A = U.A)))
3		df-ne 2019	. . . . . 6 |- (A =/= (/) <-> -. A = (/))
4	3	a1i 8	. . . . 5 |- (Ord A -> (A =/= (/) <-> -. A = (/)))
5		orduninsuc 3925	. . . . 5 |- (Ord A -> (A = U.A <-> -. E.x e. On A = suc x))
6	4, 5	anbi12d 690	. . . 4 |- (Ord A -> ((A =/= (/) /\ A = U.A) <-> (-. A = (/) /\ -. E.x e. On A = suc x)))
7		ioran 331	. . . 4 |- (-. (A = (/) \/ E.x e. On A = suc x) <-> (-. A = (/) /\ -. E.x e. On A = suc x))
8	6, 7	syl6bbr 597	. . 3 |- (Ord A -> ((A =/= (/) /\ A = U.A) <-> -. (A = (/) \/ E.x e. On A = suc x)))
9	8	pm5.32i 707	. 2 |- ((Ord A /\ (A =/= (/) /\ A = U.A)) <-> (Ord A /\ -. (A = (/) \/ E.x e. On A = suc x)))
10	1, 2, 9	3bitri 194	1 |- (Lim A <-> (Ord A /\ -. (A = (/) \/ E.x e. On A = suc x)))

Syntax hints:  -. wn 2   <-> wb 163   \/ wo 239   /\ wa 240   /\ w3a 858   = wceq 1298   =/= wne 2017  E.wrex 2106  (/)c0 2875  U.cuni 3177  Ord word 3656  Oncon0 3657  Lim wlim 3658  suc csuc 3659

This theorem is referenced by:  nlimon 3935  tfinds 3942  oalimcl 5242  omlimcl 5257

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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