Theorem onzsl 7046

Description: An ordinal number is zero, a successor ordinal, or a limit ordinal number. (Contributed by NM, 1-Oct-2003.) (Proof 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		elex 3212	. . 3 |- ( A e. On -> A e. _V )
2		eloni 5733	. . 3 |- ( A e. On -> Ord A )
3		ordzsl 7045	. . . 4 |- ( Ord A <-> ( A = (/) \/ E. x e. On A = suc x \/ Lim A ) )
4		3mix1 1230	. . . . . 6 |- ( A = (/) -> ( A = (/) \/ E. x e. On A = suc x \/ ( A e. _V /\ Lim A ) ) )
5	4	adantl 482	. . . . 5 |- ( ( A e. _V /\ A = (/) ) -> ( A = (/) \/ E. x e. On A = suc x \/ ( A e. _V /\ Lim A ) ) )
6		3mix2 1231	. . . . . 6 |- ( E. x e. On A = suc x -> ( A = (/) \/ E. x e. On A = suc x \/ ( A e. _V /\ Lim A ) ) )
7	6	adantl 482	. . . . 5 |- ( ( A e. _V /\ E. x e. On A = suc x ) -> ( A = (/) \/ E. x e. On A = suc x \/ ( A e. _V /\ Lim A ) ) )
8		3mix3 1232	. . . . 5 |- ( ( A e. _V /\ Lim A ) -> ( A = (/) \/ E. x e. On A = suc x \/ ( A e. _V /\ Lim A ) ) )
9	5, 7, 8	3jaodan 1394	. . . 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 ) ) )
10	3, 9	sylan2b 492	. . 3 |- ( ( A e. _V /\ Ord A ) -> ( A = (/) \/ E. x e. On A = suc x \/ ( A e. _V /\ Lim A ) ) )
11	1, 2, 10	syl2anc 693	. 2 |- ( A e. On -> ( A = (/) \/ E. x e. On A = suc x \/ ( A e. _V /\ Lim A ) ) )
12		0elon 5778	. . . 4 |- (/) e. On
13		eleq1 2689	. . . 4 |- ( A = (/) -> ( A e. On <-> (/) e. On ) )
14	12, 13	mpbiri 248	. . 3 |- ( A = (/) -> A e. On )
15		suceloni 7013	. . . . 5 |- ( x e. On -> suc x e. On )
16		eleq1 2689	. . . . 5 |- ( A = suc x -> ( A e. On <-> suc x e. On ) )
17	15, 16	syl5ibrcom 237	. . . 4 |- ( x e. On -> ( A = suc x -> A e. On ) )
18	17	rexlimiv 3027	. . 3 |- ( E. x e. On A = suc x -> A e. On )
19		limelon 5788	. . 3 |- ( ( A e. _V /\ Lim A ) -> A e. On )
20	14, 18, 19	3jaoi 1391	. 2 |- ( ( A = (/) \/ E. x e. On A = suc x \/ ( A e. _V /\ Lim A ) ) -> A e. On )
21	11, 20	impbii 199	1 |- ( A e. On <-> ( A = (/) \/ E. x e. On A = suc x \/ ( A e. _V /\ Lim A ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   \/ w3o 1036   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   (/)c0 3915   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:  oawordeulem  7634  r1pwss  8647  r1val1  8649  pwcfsdom  9405  winalim2  9518  rankcf  9599  dfrdg4  32058