Theorem onzsl 3928

Description: An ordinal number is zero, a successor ordinal, or a limit ordinal number. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

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

Distinct variable group:   x,A

Proof of Theorem onzsl

Step	Hyp	Ref	Expression
1		elisset 2299	. . 3 |- (A e. On -> A e. _V)
2		eloni 3667	. . 3 |- (A e. On -> Ord A)
3		3mix1 1045	. . . . . 6 |- (A = (/) -> (A = (/) \/ E.x e. On A = suc x \/ (A e. _V /\ Lim A)))
4	3	adantl 424	. . . . 5 |- ((A e. _V /\ A = (/)) -> (A = (/) \/ E.x e. On A = suc x \/ (A e. _V /\ Lim A)))
5		3mix2 1046	. . . . . 6 |- (E.x e. On A = suc x -> (A = (/) \/ E.x e. On A = suc x \/ (A e. _V /\ Lim A)))
6	5	adantl 424	. . . . 5 |- ((A e. _V /\ E.x e. On A = suc x) -> (A = (/) \/ E.x e. On A = suc x \/ (A e. _V /\ Lim A)))
7		3mix3 1047	. . . . 5 |- ((A e. _V /\ Lim A) -> (A = (/) \/ E.x e. On A = suc x \/ (A e. _V /\ Lim A)))
8	4, 6, 7	3jaodan 1163	. . . 4 |- ((A e. _V /\ (A = (/) \/ E.x e. On A = suc x \/ Lim A)) -> (A = (/) \/ E.x e. On A = suc x \/ (A e. _V /\ Lim A)))
9		ordzsl 3927	. . . 4 |- (Ord A <-> (A = (/) \/ E.x e. On A = suc x \/ Lim A))
10	8, 9	sylan2b 501	. . 3 |- ((A e. _V /\ Ord A) -> (A = (/) \/ E.x e. On A = suc x \/ (A e. _V /\ Lim A)))
11	1, 2, 10	syl11anc 524	. 2 |- (A e. On -> (A = (/) \/ E.x e. On A = suc x \/ (A e. _V /\ Lim A)))
12		0elon 3716	. . . 4 |- (/) e. On
13		eleq1 1957	. . . 4 |- (A = (/) -> (A e. On <-> (/) e. On))
14	12, 13	mpbiri 211	. . 3 |- (A = (/) -> A e. On)
15		eleq1 1957	. . . . 5 |- (A = suc x -> (A e. On <-> suc x e. On))
16		suceloni 3894	. . . . 5 |- (x e. On -> suc x e. On)
17	15, 16	syl5cbir 228	. . . 4 |- (x e. On -> (A = suc x -> A e. On))
18	17	r19.23aiv 2211	. . 3 |- (E.x e. On A = suc x -> A e. On)
19		limelon 3727	. . 3 |- ((A e. _V /\ Lim A) -> A e. On)
20	14, 18, 19	3jaoi 1160	. 2 |- ((A = (/) \/ E.x e. On A = suc x \/ (A e. _V /\ Lim A)) -> A e. On)
21	11, 20	impbii 174	1 |- (A e. On <-> (A = (/) \/ E.x e. On A = suc x \/ (A e. _V /\ Lim A)))

Syntax hints:   <-> wb 163   /\ wa 240   \/ w3o 857   = wceq 1298   e. wcel 1300  E.wrex 2106  _Vcvv 2292  (/)c0 2875  Ord word 3656  Oncon0 3657  Lim wlim 3658  suc csuc 3659

This theorem is referenced by:  oawordeulem 5236  r1val1 5769

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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