Theorem onuninsuci 3921

Description: A limit ordinal is not a successor ordinal.

Hypothesis

Ref	Expression
onssi.1	|- A e. On

Assertion

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

Distinct variable group:   x,A

Proof of Theorem onuninsuci

Step	Hyp	Ref	Expression
1		onssi.1	. . . . . . . 8 |- A e. On
2	1	onirri 3776	. . . . . . 7 |- -. A e. A
3		id 73	. . . . . . . . 9 |- (A = U.A -> A = U.A)
4		df-suc 3663	. . . . . . . . . . . . 13 |- suc x = (x u. {x})
5	4	eqeq2i 1894	. . . . . . . . . . . 12 |- (A = suc x <-> A = (x u. {x}))
6		unieq 3185	. . . . . . . . . . . 12 |- (A = (x u. {x}) -> U.A = U.(x u. {x}))
7	5, 6	sylbi 216	. . . . . . . . . . 11 |- (A = suc x -> U.A = U.(x u. {x}))
8		uniun 3196	. . . . . . . . . . . 12 |- U.(x u. {x}) = (U.x u. U.{x})
9		visset 2295	. . . . . . . . . . . . . 14 |- x e. _V
10	9	unisn 3193	. . . . . . . . . . . . 13 |- U.{x} = x
11	10	uneq2i 2752	. . . . . . . . . . . 12 |- (U.x u. U.{x}) = (U.x u. x)
12	8, 11	eqtri 1908	. . . . . . . . . . 11 |- U.(x u. {x}) = (U.x u. x)
13	7, 12	syl6eq 1944	. . . . . . . . . 10 |- (A = suc x -> U.A = (U.x u. x))
14		eleq1 1957	. . . . . . . . . . . . 13 |- (A = suc x -> (A e. On <-> suc x e. On))
15	1, 14	mpbii 210	. . . . . . . . . . . 12 |- (A = suc x -> suc x e. On)
16		tron 3681	. . . . . . . . . . . . 13 |- Tr On
17		trsuc 3752	. . . . . . . . . . . . 13 |- ((Tr On /\ suc x e. On) -> x e. On)
18	16, 17	mpan 759	. . . . . . . . . . . 12 |- (suc x e. On -> x e. On)
19		eloni 3667	. . . . . . . . . . . . . 14 |- (x e. On -> Ord x)
20		ordtr 3672	. . . . . . . . . . . . . 14 |- (Ord x -> Tr x)
21	19, 20	syl 12	. . . . . . . . . . . . 13 |- (x e. On -> Tr x)
22		df-tr 3412	. . . . . . . . . . . . 13 |- (Tr x <-> U.x C_ x)
23	21, 22	sylib 215	. . . . . . . . . . . 12 |- (x e. On -> U.x C_ x)
24	15, 18, 23	3syl 24	. . . . . . . . . . 11 |- (A = suc x -> U.x C_ x)
25		ssequn1 2775	. . . . . . . . . . 11 |- (U.x C_ x <-> (U.x u. x) = x)
26	24, 25	sylib 215	. . . . . . . . . 10 |- (A = suc x -> (U.x u. x) = x)
27	13, 26	eqtrd 1925	. . . . . . . . 9 |- (A = suc x -> U.A = x)
28	3, 27	sylan9eqr 1951	. . . . . . . 8 |- ((A = suc x /\ A = U.A) -> A = x)
29	9	sucid 3744	. . . . . . . . . 10 |- x e. suc x
30		eleq2 1958	. . . . . . . . . 10 |- (A = suc x -> (x e. A <-> x e. suc x))
31	29, 30	mpbiri 211	. . . . . . . . 9 |- (A = suc x -> x e. A)
32	31	adantr 425	. . . . . . . 8 |- ((A = suc x /\ A = U.A) -> x e. A)
33	28, 32	eqeltrd 1971	. . . . . . 7 |- ((A = suc x /\ A = U.A) -> A e. A)
34	2, 33	mto 121	. . . . . 6 |- -. (A = suc x /\ A = U.A)
35		imnan 261	. . . . . 6 |- ((A = suc x -> -. A = U.A) <-> -. (A = suc x /\ A = U.A))
36	34, 35	mpbir 207	. . . . 5 |- (A = suc x -> -. A = U.A)
37	36	a1i 8	. . . 4 |- (x e. On -> (A = suc x -> -. A = U.A))
38	37	r19.23aiv 2211	. . 3 |- (E.x e. On A = suc x -> -. A = U.A)
39		suceq 3729	. . . . . 6 |- (x = U.A -> suc x = suc U.A)
40	39	eqeq2d 1895	. . . . 5 |- (x = U.A -> (A = suc x <-> A = suc U.A))
41	40	rcla4ev 2381	. . . 4 |- ((U.A e. On /\ A = suc U.A) -> E.x e. On A = suc x)
42		onuni 3874	. . . . 5 |- (A e. On -> U.A e. On)
43	1, 42	ax-mp 7	. . . 4 |- U.A e. On
44	1	onuniorsuci 3920	. . . . 5 |- (A = U.A \/ A = suc U.A)
45	44	ori 247	. . . 4 |- (-. A = U.A -> A = suc U.A)
46	41, 43, 45	sylancr 526	. . 3 |- (-. A = U.A -> E.x e. On A = suc x)
47	38, 46	impbii 174	. 2 |- (E.x e. On A = suc x <-> -. A = U.A)
48	47	con2bii 238	1 |- (A = U.A <-> -. E.x e. On A = suc x)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wrex 2106   u. cun 2591   C_ wss 2593  {csn 3044  U.cuni 3177  Tr wtr 3411  Ord word 3656  Oncon0 3657  suc csuc 3659

This theorem is referenced by:  orduninsuc 3925

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-suc 3663

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