Theorem onuninsuc 2356

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(s):   x,A

Proof of Theorem onuninsuc

Step	Hyp	Ref	Expression
1		on.1	. . . . . . . 8 |- A e. On
2	1	oneirr 2345	. . . . . . 7 |- -. A e. A
3		id 9	. . . . . . . . 9 |- (A = U.A -> A = U.A)
4		df-suc 2205	. . . . . . . . . . . . 13 |- suc x = (x u. {x})
5	4	cleq2i 1111	. . . . . . . . . . . 12 |- (A = suc x <-> A = (x u. {x}))
6		unieq 1927	. . . . . . . . . . . 12 |- (A = (x u. {x}) -> U.A = U.(x u. {x}))
7	5, 6	sylbi 174	. . . . . . . . . . 11 |- (A = suc x -> U.A = U.(x u. {x}))
8		uniun 1934	. . . . . . . . . . . 12 |- U.(x u. {x}) = (U.x u. U.{x})
9		visset 1350	. . . . . . . . . . . . . 14 |- x e. V
10	9	unisn 1932	. . . . . . . . . . . . 13 |- U.{x} = x
11	10	uneq2i 1608	. . . . . . . . . . . 12 |- (U.x u. U.{x}) = (U.x u. x)
12	8, 11	eqtr 1119	. . . . . . . . . . 11 |- U.(x u. {x}) = (U.x u. x)
13	7, 12	syl6eq 1140	. . . . . . . . . 10 |- (A = suc x -> U.A = (U.x u. x))
14		eleq1 1149	. . . . . . . . . . . . 13 |- (A = suc x -> (A e. On <-> suc x e. On))
15	1, 14	mpbii 168	. . . . . . . . . . . 12 |- (A = suc x -> suc x e. On)
16		ordon 2238	. . . . . . . . . . . . . 14 |- Ord On
17		ordtr 2213	. . . . . . . . . . . . . 14 |- (Ord On -> Tr On)
18	16, 17	ax-mp 6	. . . . . . . . . . . . 13 |- Tr On
19		trsuc 2308	. . . . . . . . . . . . 13 |- ((Tr On /\ suc x e. On) -> x e. On)
20	18, 19	mpan 518	. . . . . . . . . . . 12 |- (suc x e. On -> x e. On)
21		eloni 2209	. . . . . . . . . . . . . 14 |- (x e. On -> Ord x)
22		ordtr 2213	. . . . . . . . . . . . . 14 |- (Ord x -> Tr x)
23	21, 22	syl 12	. . . . . . . . . . . . 13 |- (x e. On -> Tr x)
24		df-tr 2042	. . . . . . . . . . . . 13 |- (Tr x <-> U.x (_ x)
25	23, 24	sylib 173	. . . . . . . . . . . 12 |- (x e. On -> U.x (_ x)
26	15, 20, 25	3syl 21	. . . . . . . . . . 11 |- (A = suc x -> U.x (_ x)
27		ssequn1 1628	. . . . . . . . . . 11 |- (U.x (_ x <-> (U.x u. x) = x)
28	26, 27	sylib 173	. . . . . . . . . 10 |- (A = suc x -> (U.x u. x) = x)
29	13, 28	eqtrd 1128	. . . . . . . . 9 |- (A = suc x -> U.A = x)
30	3, 29	sylan9eqr 1145	. . . . . . . 8 |- ((A = suc x /\ A = U.A) -> A = x)
31	9	sucid 2304	. . . . . . . . . 10 |- x e. suc x
32		eleq2 1150	. . . . . . . . . 10 |- (A = suc x -> (x e. A <-> x e. suc x))
33	31, 32	mpbiri 169	. . . . . . . . 9 |- (A = suc x -> x e. A)
34	33	adantr 306	. . . . . . . 8 |- ((A = suc x /\ A = U.A) -> x e. A)
35	30, 34	eqeltrd 1163	. . . . . . 7 |- ((A = suc x /\ A = U.A) -> A e. A)
36	2, 35	mto 93	. . . . . 6 |- -. (A = suc x /\ A = U.A)
37		imnan 207	. . . . . 6 |- ((A = suc x -> -. A = U.A) <-> -. (A = suc x /\ A = U.A))
38	36, 37	mpbir 165	. . . . 5 |- (A = suc x -> -. A = U.A)
39	38	a1i 7	. . . 4 |- (x e. On -> (A = suc x -> -. A = U.A))
40	39	r19.23aiv 1284	. . 3 |- (E.x e. On A = suc x -> -. A = U.A)
41	1	onuniorsuc 2355	. . . . . 6 |- (A = U.A \/ A = suc U.A)
42	41	ori 200	. . . . 5 |- (-. A = U.A -> A = suc U.A)
43	1	onss 2347	. . . . . 6 |- A (_ On
44	1	elisseti 1355	. . . . . . 7 |- A e. V
45	44	onuni 2251	. . . . . 6 |- (A (_ On -> U.A e. On)
46	43, 45	ax-mp 6	. . . . 5 |- U.A e. On
47	42, 46	jctil 240	. . . 4 |- (-. A = U.A -> (U.A e. On /\ A = suc U.A))
48		suceq 2288	. . . . . 6 |- (x = U.A -> suc x = suc U.A)
49	48	cleq2d 1112	. . . . 5 |- (x = U.A -> (A = suc x <-> A = suc U.A))
50	49	rcla4ev 1403	. . . 4 |- ((U.A e. On /\ A = suc U.A) -> E.x e. On A = suc x)
51	47, 50	syl 12	. . 3 |- (-. A = U.A -> E.x e. On A = suc x)
52	40, 51	impbi 139	. 2 |- (E.x e. On A = suc x <-> -. A = U.A)
53	52	bicon2i 194	1 |- (A = U.A <-> -. E.x e. On A = suc x)

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092  E.wrex 1202   u. cun 1485   (_ wss 1487  {csn 1808  U.cuni 1919  Tr wtr 2041  Ord word 2198  Oncon0 2199  suc csuc 2201

This theorem is referenced by:  orduninsuc 2365

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-suc 2205

metamath.org