Theorem onuninsuc 3114

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 3103	. . . . . . 7 |- -. A e. A
3		id 59	. . . . . . . . 9 |- (A = U.A -> A = U.A)
4		df-suc 2960	. . . . . . . . . . . . 13 |- suc x = (x u. {x})
5	4	eqeq2i 1488	. . . . . . . . . . . 12 |- (A = suc x <-> A = (x u. {x}))
6		unieq 2514	. . . . . . . . . . . 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 2523	. . . . . . . . . . . 12 |- U.(x u. {x}) = (U.x u. U.{x})
9		visset 1816	. . . . . . . . . . . . . 14 |- x e. V
10	9	unisn 2521	. . . . . . . . . . . . 13 |- U.{x} = x
11	10	uneq2i 2184	. . . . . . . . . . . 12 |- (U.x u. U.{x}) = (U.x u. x)
12	8, 11	eqtr 1498	. . . . . . . . . . 11 |- U.(x u. {x}) = (U.x u. x)
13	7, 12	syl6eq 1526	. . . . . . . . . 10 |- (A = suc x -> U.A = (U.x u. x))
14		eleq1 1537	. . . . . . . . . . . . 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 2993	. . . . . . . . . . . . . 14 |- Ord On
17		ordtr 2968	. . . . . . . . . . . . . 14 |- (Ord On -> Tr On)
18	16, 17	ax-mp 7	. . . . . . . . . . . . 13 |- Tr On
19		trsuc 3061	. . . . . . . . . . . . 13 |- ((Tr On /\ suc x e. On) -> x e. On)
20	18, 19	mpan 697	. . . . . . . . . . . 12 |- (suc x e. On -> x e. On)
21		eloni 2964	. . . . . . . . . . . . . 14 |- (x e. On -> Ord x)
22		ordtr 2968	. . . . . . . . . . . . . 14 |- (Ord x -> Tr x)
23	21, 22	syl 10	. . . . . . . . . . . . 13 |- (x e. On -> Tr x)
24		df-tr 2686	. . . . . . . . . . . . 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 2203	. . . . . . . . . . 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 1510	. . . . . . . . 9 |- (A = suc x -> U.A = x)
30	3, 29	sylan9eqr 1532	. . . . . . . 8 |- ((A = suc x /\ A = U.A) -> A = x)
31	9	sucid 3057	. . . . . . . . . 10 |- x e. suc x
32		eleq2 1538	. . . . . . . . . 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 391	. . . . . . . 8 |- ((A = suc x /\ A = U.A) -> x e. A)
35	30, 34	eqeltrd 1551	. . . . . . 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 1746	. . 3 |- (E.x e. On A = suc x -> -. A = U.A)
41	1	onuniorsuc 3113	. . . . . 6 |- (A = U.A \/ A = suc U.A)
42	41	ori 230	. . . . 5 |- (-. A = U.A -> A = suc U.A)
43		onuni 3002	. . . . . 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 3040	. . . . . 6 |- (x = U.A -> suc x = suc U.A)
47	46	eqeq2d 1489	. . . . 5 |- (x = U.A -> (A = suc x <-> A = suc U.A))
48	47	rcla4ev 1880	. . . 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 958   e. wcel 960  E.wrex 1649   u. cun 2048   (_ wss 2050  {csn 2413  U.cuni 2507  Tr wtr 2685  Ord word 2953  Oncon0 2954  suc csuc 2956

This theorem is referenced by:  orduninsuc 3120

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-suc 2960

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