Theorem ordzsl 3106

Description: An ordinal is zero, a successor ordinal, or a limit ordinal.

Assertion

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

Distinct variable group:   x,A

Proof of Theorem ordzsl

Step	Hyp	Ref	Expression
1		orduninsuc 3104	. . . . . 6 |- (Ord A -> (A = U.A <-> -. E.x e. On A = suc x))
2	1	biimprd 154	. . . . 5 |- (Ord A -> (-. E.x e. On A = suc x -> A = U.A))
3		unizlim 3103	. . . . 5 |- (Ord A -> (A = U.A <-> (A = (/) \/ Lim A)))
4	2, 3	sylibd 202	. . . 4 |- (Ord A -> (-. E.x e. On A = suc x -> (A = (/) \/ Lim A)))
5	4	orrd 233	. . 3 |- (Ord A -> (E.x e. On A = suc x \/ (A = (/) \/ Lim A)))
6		3orass 776	. . . 4 |- ((A = (/) \/ E.x e. On A = suc x \/ Lim A) <-> (A = (/) \/ (E.x e. On A = suc x \/ Lim A)))
7		or12 258	. . . 4 |- ((A = (/) \/ (E.x e. On A = suc x \/ Lim A)) <-> (E.x e. On A = suc x \/ (A = (/) \/ Lim A)))
8	6, 7	bitr 173	. . 3 |- ((A = (/) \/ E.x e. On A = suc x \/ Lim A) <-> (E.x e. On A = suc x \/ (A = (/) \/ Lim A)))
9	5, 8	sylibr 200	. 2 |- (Ord A -> (A = (/) \/ E.x e. On A = suc x \/ Lim A))
10		ord0 3011	. . . 4 |- Ord (/)
11		ordeq 2945	. . . 4 |- (A = (/) -> (Ord A <-> Ord (/)))
12	10, 11	mpbiri 194	. . 3 |- (A = (/) -> Ord A)
13		eleq1 1526	. . . . . 6 |- (A = suc x -> (A e. On <-> suc x e. On))
14		suceloni 3052	. . . . . 6 |- (x e. On -> suc x e. On)
15	13, 14	syl5bir 210	. . . . 5 |- (A = suc x -> (x e. On -> A e. On))
16		eloni 2948	. . . . 5 |- (A e. On -> Ord A)
17	15, 16	syl6com 53	. . . 4 |- (x e. On -> (A = suc x -> Ord A))
18	17	r19.23aiv 1735	. . 3 |- (E.x e. On A = suc x -> Ord A)
19		limord 3018	. . 3 |- (Lim A -> Ord A)
20	12, 18, 19	3jaoi 884	. 2 |- ((A = (/) \/ E.x e. On A = suc x \/ Lim A) -> Ord A)
21	9, 20	impbi 157	1 |- (Ord A <-> (A = (/) \/ E.x e. On A = suc x \/ Lim A))

Syntax hints:  -. wn 2   <-> wb 146   \/ wo 222   \/ w3o 772   = wceq 953   e. wcel 955  E.wrex 1638  (/)c0 2270  U.cuni 2493  Ord word 2937  Oncon0 2938  Lim wlim 2939  suc csuc 2940

This theorem is referenced by:  onzsl 3107  dflim3 3108  rankr1 4646  rankxplim3 4686  rankxpsuc 4687  cardlim 4823  cardaleph 4857

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

Copyright terms: Public domain