Theorem ordzsl 7045

Description: An ordinal is zero, a successor ordinal, or a limit ordinal. (Contributed by NM, 1-Oct-2003.)

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 7043	. . . . . 6 |- ( Ord A -> ( A = U. A <-> -. E. x e. On A = suc x ) )
2	1	biimprd 238	. . . . 5 |- ( Ord A -> ( -. E. x e. On A = suc x -> A = U. A ) )
3		unizlim 5844	. . . . 5 |- ( Ord A -> ( A = U. A <-> ( A = (/) \/ Lim A ) ) )
4	2, 3	sylibd 229	. . . 4 |- ( Ord A -> ( -. E. x e. On A = suc x -> ( A = (/) \/ Lim A ) ) )
5	4	orrd 393	. . 3 |- ( Ord A -> ( E. x e. On A = suc x \/ ( A = (/) \/ Lim A ) ) )
6		3orass 1040	. . . 4 |- ( ( A = (/) \/ E. x e. On A = suc x \/ Lim A ) <-> ( A = (/) \/ ( E. x e. On A = suc x \/ Lim A ) ) )
7		or12 545	. . . 4 |- ( ( A = (/) \/ ( E. x e. On A = suc x \/ Lim A ) ) <-> ( E. x e. On A = suc x \/ ( A = (/) \/ Lim A ) ) )
8	6, 7	bitri 264	. . 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 224	. 2 |- ( Ord A -> ( A = (/) \/ E. x e. On A = suc x \/ Lim A ) )
10		ord0 5777	. . . 4 |- Ord (/)
11		ordeq 5730	. . . 4 |- ( A = (/) -> ( Ord A <-> Ord (/) ) )
12	10, 11	mpbiri 248	. . 3 |- ( A = (/) -> Ord A )
13		suceloni 7013	. . . . . 6 |- ( x e. On -> suc x e. On )
14		eleq1 2689	. . . . . 6 |- ( A = suc x -> ( A e. On <-> suc x e. On ) )
15	13, 14	syl5ibr 236	. . . . 5 |- ( A = suc x -> ( x e. On -> A e. On ) )
16		eloni 5733	. . . . 5 |- ( A e. On -> Ord A )
17	15, 16	syl6com 37	. . . 4 |- ( x e. On -> ( A = suc x -> Ord A ) )
18	17	rexlimiv 3027	. . 3 |- ( E. x e. On A = suc x -> Ord A )
19		limord 5784	. . 3 |- ( Lim A -> Ord A )
20	12, 18, 19	3jaoi 1391	. 2 |- ( ( A = (/) \/ E. x e. On A = suc x \/ Lim A ) -> Ord A )
21	9, 20	impbii 199	1 |- ( Ord A <-> ( A = (/) \/ E. x e. On A = suc x \/ Lim A ) )

Syntax hints:   -. wn 3   <-> wb 196   \/ wo 383   \/ w3o 1036   = wceq 1483   e. wcel 1990   E.wrex 2913   (/)c0 3915   U.cuni 4436   Ord word 5722   Oncon0 5723   Lim wlim 5724   suc csuc 5725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729

This theorem is referenced by:  onzsl  7046  tfrlem16  7489  omeulem1  7662  oaabs2  7725  rankxplim3  8744  rankxpsuc  8745  cardlim  8798  cardaleph  8912  cflim2  9085  dfrdg2  31701