Theorem ordzsl 3122

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 3120	. . . . . 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 3119	. . . . 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 780	. . . 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 3027	. . . 4 |- Ord (/)
11		ordeq 2961	. . . 4 |- (A = (/) -> (Ord A <-> Ord (/)))
12	10, 11	mpbiri 194	. . 3 |- (A = (/) -> Ord A)
13		eleq1 1537	. . . . . 6 |- (A = suc x -> (A e. On <-> suc x e. On))
14		suceloni 3068	. . . . . 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 2964	. . . . 5 |- (A e. On -> Ord A)
17	15, 16	syl6com 53	. . . 4 |- (x e. On -> (A = suc x -> Ord A))
18	17	r19.23aiv 1746	. . 3 |- (E.x e. On A = suc x -> Ord A)
19		limord 3034	. . 3 |- (Lim A -> Ord A)
20	12, 18, 19	3jaoi 889	. 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 776   = wceq 958   e. wcel 960  E.wrex 1649  (/)c0 2283  U.cuni 2507  Ord word 2953  Oncon0 2954  Lim wlim 2955  suc csuc 2956

This theorem is referenced by:  onzsl 3123  dflim3 3124  rankr1 4684  rankxplim3 4724  rankxpsuc 4725  cardlim 4862  cardaleph 4896

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960

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