Theorem onuninsuc 3098

Description: A limit ordinal is not a successor ordinal.

Hypothesis

Ref	Expression
on.1	|- A e. On

Assertion

Ref	Expression
onuninsuc	|- (A = U.A <-> -. E.x e. On A = suc x)

Distinct variable group:   x,A

Proof of Theorem onuninsuc

Step	Hyp	Ref	Expression
1		on.1	. . . . . . . 8 |- A e. On
2	1	onirr 3087	. . . . . . 7 |- -. A e. A
3		id 59	. . . . . . . . 9 |- (A = U.A -> A = U.A)
4		df-suc 2944	. . . . . . . . . . . . 13 |- suc x = (x u. {x})
5	4	eqeq2i 1477	. . . . . . . . . . . 12 |- (A = suc x <-> A = (x u. {x}))
6		unieq 2500	. . . . . . . . . . . 12 |- (A = (x u. {x}) -> U.A = U.(x u. {x}))
7	5, 6	sylbi 199	. . . . . . . . . . 11 |- (A = suc x -> U.A = U.(x u. {x}))
8		uniun 2509	. . . . . . . . . . . 12 |- U.(x u. {x}) = (U.x u. U.{x})
9		visset 1804	. . . . . . . . . . . . . 14 |- x e. V
10	9	unisn 2507	. . . . . . . . . . . . 13 |- U.{x} = x
11	10	uneq2i 2171	. . . . . . . . . . . 12 |- (U.x u. U.{x}) = (U.x u. x)
12	8, 11	eqtr 1487	. . . . . . . . . . 11 |- U.(x u. {x}) = (U.x u. x)
13	7, 12	syl6eq 1515	. . . . . . . . . 10 |- (A = suc x -> U.A = (U.x u. x))
14		eleq1 1526	. . . . . . . . . . . . 13 |- (A = suc x -> (A e. On <-> suc x e. On))
15	1, 14	mpbii 193	. . . . . . . . . . . 12 |- (A = suc x -> suc x e. On)
16		ordon 2977	. . . . . . . . . . . . . 14 |- Ord On
17		ordtr 2952	. . . . . . . . . . . . . 14 |- (Ord On -> Tr On)
18	16, 17	ax-mp 7	. . . . . . . . . . . . 13 |- Tr On
19		trsuc 3045	. . . . . . . . . . . . 13 |- ((Tr On /\ suc x e. On) -> x e. On)
20	18, 19	mpan 693	. . . . . . . . . . . 12 |- (suc x e. On -> x e. On)
21		eloni 2948	. . . . . . . . . . . . . 14 |- (x e. On -> Ord x)
22		ordtr 2952	. . . . . . . . . . . . . 14 |- (Ord x -> Tr x)
23	21, 22	syl 10	. . . . . . . . . . . . 13 |- (x e. On -> Tr x)
24		df-tr 2671	. . . . . . . . . . . . 13 |- (Tr x <-> U.x (_ x)
25	23, 24	sylib 198	. . . . . . . . . . . 12 |- (x e. On -> U.x (_ x)
26	15, 20, 25	3syl 20	. . . . . . . . . . 11 |- (A = suc x -> U.x (_ x)
27		ssequn1 2190	. . . . . . . . . . 11 |- (U.x (_ x <-> (U.x u. x) = x)
28	26, 27	sylib 198	. . . . . . . . . 10 |- (A = suc x -> (U.x u. x) = x)
29	13, 28	eqtrd 1499	. . . . . . . . 9 |- (A = suc x -> U.A = x)
30	3, 29	sylan9eqr 1521	. . . . . . . 8 |- ((A = suc x /\ A = U.A) -> A = x)
31	9	sucid 3041	. . . . . . . . . 10 |- x e. suc x
32		eleq2 1527	. . . . . . . . . 10 |- (A = suc x -> (x e. A <-> x e. suc x))
33	31, 32	mpbiri 194	. . . . . . . . 9 |- (A = suc x -> x e. A)
34	33	adantr 389	. . . . . . . 8 |- ((A = suc x /\ A = U.A) -> x e. A)
35	30, 34	eqeltrd 1540	. . . . . . 7 |- ((A = suc x /\ A = U.A) -> A e. A)
36	2, 35	mto 106	. . . . . 6 |- -. (A = suc x /\ A = U.A)
37		imnan 242	. . . . . 6 |- ((A = suc x -> -. A = U.A) <-> -. (A = suc x /\ A = U.A))
38	36, 37	mpbir 190	. . . . 5 |- (A = suc x -> -. A = U.A)
39	38	a1i 8	. . . 4 |- (x e. On -> (A = suc x -> -. A = U.A))
40	39	r19.23aiv 1735	. . 3 |- (E.x e. On A = suc x -> -. A = U.A)
41	1	onuniorsuc 3097	. . . . . 6 |- (A = U.A \/ A = suc U.A)
42	41	ori 230	. . . . 5 |- (-. A = U.A -> A = suc U.A)
43		onuni 2986	. . . . . 6 |- (A e. On -> U.A e. On)
44	1, 43	ax-mp 7	. . . . 5 |- U.A e. On
45	42, 44	jctil 292	. . . 4 |- (-. A = U.A -> (U.A e. On /\ A = suc U.A))
46		suceq 3024	. . . . . 6 |- (x = U.A -> suc x = suc U.A)
47	46	eqeq2d 1478	. . . . 5 |- (x = U.A -> (A = suc x <-> A = suc U.A))
48	47	rcla4ev 1868	. . . 4 |- ((U.A e. On /\ A = suc U.A) -> E.x e. On A = suc x)
49	45, 48	syl 10	. . 3 |- (-. A = U.A -> E.x e. On A = suc x)
50	40, 49	impbi 157	. 2 |- (E.x e. On A = suc x <-> -. A = U.A)
51	50	con2bii 221	1 |- (A = U.A <-> -. E.x e. On A = suc x)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  E.wrex 1638   u. cun 2035   (_ wss 2037  {csn 2399  U.cuni 2493  Tr wtr 2670  Ord word 2937  Oncon0 2938  suc csuc 2940

This theorem is referenced by:  orduninsuc 3104

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-suc 2944

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