Theorem onzsl 3174

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

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 1864	. . 3 |- (A e. On -> A e. V)
2	1	pm4.71ri 649	. 2 |- (A e. On <-> (A e. V /\ A e. On))
3		elong 3013	. . 3 |- (A e. V -> (A e. On <-> Ord A))
4	3	pm5.32i 656	. 2 |- ((A e. V /\ A e. On) <-> (A e. V /\ Ord A))
5		andi 615	. . . 4 |- ((A e. V /\ ((A = (/) \/ E.x e. On A = suc x) \/ Lim A)) <-> ((A e. V /\ (A = (/) \/ E.x e. On A = suc x)) \/ (A e. V /\ Lim A)))
6		0ex 2766	. . . . . . . 8 |- (/) e. V
7		eleq1 1581	. . . . . . . 8 |- (A = (/) -> (A e. V <-> (/) e. V))
8	6, 7	mpbiri 201	. . . . . . 7 |- (A = (/) -> A e. V)
9		visset 1860	. . . . . . . . . . 11 |- x e. V
10	9	sucex 3107	. . . . . . . . . 10 |- suc x e. V
11		eleq1 1581	. . . . . . . . . 10 |- (A = suc x -> (A e. V <-> suc x e. V))
12	10, 11	mpbiri 201	. . . . . . . . 9 |- (A = suc x -> A e. V)
13	12	a1i 8	. . . . . . . 8 |- (x e. On -> (A = suc x -> A e. V))
14	13	r19.23aiv 1790	. . . . . . 7 |- (E.x e. On A = suc x -> A e. V)
15	8, 14	jaoi 348	. . . . . 6 |- ((A = (/) \/ E.x e. On A = suc x) -> A e. V)
16	15	pm4.71ri 649	. . . . 5 |- ((A = (/) \/ E.x e. On A = suc x) <-> (A e. V /\ (A = (/) \/ E.x e. On A = suc x)))
17	16	orbi1i 263	. . . 4 |- (((A = (/) \/ E.x e. On A = suc x) \/ (A e. V /\ Lim A)) <-> ((A e. V /\ (A = (/) \/ E.x e. On A = suc x)) \/ (A e. V /\ Lim A)))
18	5, 17	bitr4i 183	. . 3 |- ((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)))
19		ordzsl 3173	. . . . 5 |- (Ord A <-> (A = (/) \/ E.x e. On A = suc x \/ Lim A))
20		df-3or 788	. . . . 5 |- ((A = (/) \/ E.x e. On A = suc x \/ Lim A) <-> ((A = (/) \/ E.x e. On A = suc x) \/ Lim A))
21	19, 20	bitri 180	. . . 4 |- (Ord A <-> ((A = (/) \/ E.x e. On A = suc x) \/ Lim A))
22	21	anbi2i 491	. . 3 |- ((A e. V /\ Ord A) <-> (A e. V /\ ((A = (/) \/ E.x e. On A = suc x) \/ Lim A)))
23		df-3or 788	. . 3 |- ((A = (/) \/ E.x e. On A = suc x \/ (A e. V /\ Lim A)) <-> ((A = (/) \/ E.x e. On A = suc x) \/ (A e. V /\ Lim A)))
24	18, 22, 23	3bitr4i 190	. 2 |- ((A e. V /\ Ord A) <-> (A = (/) \/ E.x e. On A = suc x \/ (A e. V /\ Lim A)))
25	2, 4, 24	3bitri 184	1 |- (A e. On <-> (A = (/) \/ E.x e. On A = suc x \/ (A e. V /\ Lim A)))

Syntax hints:   -> wi 3   <-> wb 153   \/ wo 229   /\ wa 230   \/ w3o 786   = wceq 997   e. wcel 999  E.wrex 1693  Vcvv 1858  (/)c0 2331  Ord word 3004  Oncon0 3005  Lim wlim 3006  suc csuc 3007

This theorem is referenced by:  oawordeulem 4246  r1val1 4720

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-10 1007  ax-11 1008  ax-12 1009  ax-13 1010  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504  ax-sep 2758  ax-nul 2765  ax-pow 2798  ax-pr 2835  ax-un 2922

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-3or 788  df-3an 789  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-ral 1696  df-rex 1697  df-v 1859  df-dif 2100  df-un 2101  df-in 2102  df-ss 2104  df-nul 2332  df-if 2414  df-pw 2454  df-sn 2464  df-pr 2465  df-tp 2467  df-op 2468  df-uni 2558  df-br 2675  df-opab 2722  df-tr 2736  df-eprel 2888  df-po 2896  df-so 2906  df-fr 2974  df-we 2991  df-ord 3008  df-on 3009  df-lim 3010  df-suc 3011

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