Theorem onuninsuci 3194

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 3076	. . . . . . 7 |- -. A e. A
3		id 59	. . . . . . . . 9 |- (A = U.A -> A = U.A)
4		df-suc 2981	. . . . . . . . . . . . 13 |- suc x = (x u. {x})
5	4	eqeq2i 1528	. . . . . . . . . . . 12 |- (A = suc x <-> A = (x u. {x}))
6		unieq 2576	. . . . . . . . . . . 12 |- (A = (x u. {x}) -> U.A = U.(x u. {x}))
7	5, 6	sylbi 197	. . . . . . . . . . 11 |- (A = suc x -> U.A = U.(x u. {x}))
8		uniun 2586	. . . . . . . . . . . 12 |- U.(x u. {x}) = (U.x u. U.{x})
9		visset 1859	. . . . . . . . . . . . . 14 |- x e. V
10	9	unisn 2583	. . . . . . . . . . . . 13 |- U.{x} = x
11	10	uneq2i 2233	. . . . . . . . . . . 12 |- (U.x u. U.{x}) = (U.x u. x)
12	8, 11	eqtri 1538	. . . . . . . . . . 11 |- U.(x u. {x}) = (U.x u. x)
13	7, 12	syl6eq 1566	. . . . . . . . . 10 |- (A = suc x -> U.A = (U.x u. x))
14		eleq1 1577	. . . . . . . . . . . . 13 |- (A = suc x -> (A e. On <-> suc x e. On))
15	1, 14	mpbii 191	. . . . . . . . . . . 12 |- (A = suc x -> suc x e. On)
16		tron 2998	. . . . . . . . . . . . 13 |- Tr On
17		trsuc 3056	. . . . . . . . . . . . 13 |- ((Tr On /\ suc x e. On) -> x e. On)
18	16, 17	mpan 699	. . . . . . . . . . . 12 |- (suc x e. On -> x e. On)
19		eloni 2985	. . . . . . . . . . . . . 14 |- (x e. On -> Ord x)
20		ordtr 2989	. . . . . . . . . . . . . 14 |- (Ord x -> Tr x)
21	19, 20	syl 10	. . . . . . . . . . . . 13 |- (x e. On -> Tr x)
22		df-tr 2755	. . . . . . . . . . . . 13 |- (Tr x <-> U.x (_ x)
23	21, 22	sylib 196	. . . . . . . . . . . 12 |- (x e. On -> U.x (_ x)
24	15, 18, 23	3syl 20	. . . . . . . . . . 11 |- (A = suc x -> U.x (_ x)
25		ssequn1 2252	. . . . . . . . . . 11 |- (U.x (_ x <-> (U.x u. x) = x)
26	24, 25	sylib 196	. . . . . . . . . 10 |- (A = suc x -> (U.x u. x) = x)
27	13, 26	eqtrd 1550	. . . . . . . . 9 |- (A = suc x -> U.A = x)
28	3, 27	sylan9eqr 1572	. . . . . . . 8 |- ((A = suc x /\ A = U.A) -> A = x)
29	9	sucid 3051	. . . . . . . . . 10 |- x e. suc x
30		eleq2 1578	. . . . . . . . . 10 |- (A = suc x -> (x e. A <-> x e. suc x))
31	29, 30	mpbiri 192	. . . . . . . . 9 |- (A = suc x -> x e. A)
32	31	adantr 389	. . . . . . . 8 |- ((A = suc x /\ A = U.A) -> x e. A)
33	28, 32	eqeltrd 1591	. . . . . . 7 |- ((A = suc x /\ A = U.A) -> A e. A)
34	2, 33	mto 105	. . . . . 6 |- -. (A = suc x /\ A = U.A)
35		imnan 240	. . . . . 6 |- ((A = suc x -> -. A = U.A) <-> -. (A = suc x /\ A = U.A))
36	34, 35	mpbir 188	. . . . 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 1789	. . 3 |- (E.x e. On A = suc x -> -. A = U.A)
39	1	onuniorsuci 3193	. . . . . 6 |- (A = U.A \/ A = suc U.A)
40	39	ori 228	. . . . 5 |- (-. A = U.A -> A = suc U.A)
41		onuni 3150	. . . . . 6 |- (A e. On -> U.A e. On)
42	1, 41	ax-mp 7	. . . . 5 |- U.A e. On
43	40, 42	jctil 290	. . . 4 |- (-. A = U.A -> (U.A e. On /\ A = suc U.A))
44		suceq 3038	. . . . . 6 |- (x = U.A -> suc x = suc U.A)
45	44	eqeq2d 1529	. . . . 5 |- (x = U.A -> (A = suc x <-> A = suc U.A))
46	45	rcla4ev 1923	. . . 4 |- ((U.A e. On /\ A = suc U.A) -> E.x e. On A = suc x)
47	43, 46	syl 10	. . 3 |- (-. A = U.A -> E.x e. On A = suc x)
48	38, 47	impbii 155	. 2 |- (E.x e. On A = suc x <-> -. A = U.A)
49	48	con2bii 219	1 |- (A = U.A <-> -. E.x e. On A = suc x)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 144   /\ wa 221   = wceq 992   e. wcel 994  E.wrex 1692   u. cun 2097   (_ wss 2099  {csn 2467  U.cuni 2569  Tr wtr 2754  Ord word 2974  Oncon0 2975  suc csuc 2977

This theorem is referenced by:  orduninsuc 3197

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-sep 2777  ax-pow 2818  ax-pr 2855  ax-un 3089

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-v 1858  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-suc 2981

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