Theorem onzsl 3117

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 1817	. . 3 |- (A e. On -> A e. V)
2	1	pm4.71ri 638	. 2 |- (A e. On <-> (A e. V /\ A e. On))
3		elong 2956	. . 3 |- (A e. V -> (A e. On <-> Ord A))
4	3	pm5.32i 645	. 2 |- ((A e. V /\ A e. On) <-> (A e. V /\ Ord A))
5		andi 604	. . . 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 2711	. . . . . . . 8 |- (/) e. V
7		eleq1 1534	. . . . . . . 8 |- (A = (/) -> (A e. V <-> (/) e. V))
8	6, 7	mpbiri 194	. . . . . . 7 |- (A = (/) -> A e. V)
9		visset 1813	. . . . . . . . . . 11 |- x e. V
10	9	sucex 3050	. . . . . . . . . 10 |- suc x e. V
11		eleq1 1534	. . . . . . . . . 10 |- (A = suc x -> (A e. V <-> suc x e. V))
12	10, 11	mpbiri 194	. . . . . . . . 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 1743	. . . . . . 7 |- (E.x e. On A = suc x -> A e. V)
15	8, 14	jaoi 341	. . . . . 6 |- ((A = (/) \/ E.x e. On A = suc x) -> A e. V)
16	15	pm4.71ri 638	. . . . 5 |- ((A = (/) \/ E.x e. On A = suc x) <-> (A e. V /\ (A = (/) \/ E.x e. On A = suc x)))
17	16	orbi1i 256	. . . 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	bitr4 176	. . 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 3116	. . . . 5 |- (Ord A <-> (A = (/) \/ E.x e. On A = suc x \/ Lim A))
20		df-3or 776	. . . . 5 |- ((A = (/) \/ E.x e. On A = suc x \/ Lim A) <-> ((A = (/) \/ E.x e. On A = suc x) \/ Lim A))
21	19, 20	bitr 173	. . . 4 |- (Ord A <-> ((A = (/) \/ E.x e. On A = suc x) \/ Lim A))
22	21	anbi2i 480	. . 3 |- ((A e. V /\ Ord A) <-> (A e. V /\ ((A = (/) \/ E.x e. On A = suc x) \/ Lim A)))
23		df-3or 776	. . 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	3bitr4 183	. 2 |- ((A e. V /\ Ord A) <-> (A = (/) \/ E.x e. On A = suc x \/ (A e. V /\ Lim A)))
25	2, 4, 24	3bitr 177	1 |- (A e. On <-> (A = (/) \/ E.x e. On A = suc x \/ (A e. V /\ Lim A)))

Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   \/ w3o 774   = wceq 956   e. wcel 958  E.wrex 1646  Vcvv 1811  (/)c0 2280  Ord word 2947  Oncon0 2948  Lim wlim 2949  suc csuc 2950

This theorem is referenced by:  oawordeulem 4188  r1val1 4658

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954

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