Theorem orduninsuc 2365

Description: A limit ordinal is not a successor ordinal.

Assertion

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

Distinct variable group(s):   x,A

Proof of Theorem orduninsuc

Step	Hyp	Ref	Expression
1		ordeleqon 2241	. 2 |- (Ord A <-> (A e. On \/ A = On))
2		id 9	. . . . . 6 |- (A = if(A e. On, A, (/)) -> A = if(A e. On, A, (/)))
3		unieq 1927	. . . . . 6 |- (A = if(A e. On, A, (/)) -> U.A = U.if(A e. On, A, (/)))
4	2, 3	cleq12d 1115	. . . . 5 |- (A = if(A e. On, A, (/)) -> (A = U.A <-> if(A e. On, A, (/)) = U.if(A e. On, A, (/))))
5		cleq1 1107	. . . . . . 7 |- (A = if(A e. On, A, (/)) -> (A = suc x <-> if(A e. On, A, (/)) = suc x))
6	5	birexdv 1220	. . . . . 6 |- (A = if(A e. On, A, (/)) -> (E.x e. On A = suc x <-> E.x e. On if(A e. On, A, (/)) = suc x))
7	6	negbid 463	. . . . 5 |- (A = if(A e. On, A, (/)) -> (-. E.x e. On A = suc x <-> -. E.x e. On if(A e. On, A, (/)) = suc x))
8	4, 7	bibi12d 477	. . . 4 |- (A = if(A e. On, A, (/)) -> ((A = U.A <-> -. E.x e. On A = suc x) <-> (if(A e. On, A, (/)) = U.if(A e. On, A, (/)) <-> -. E.x e. On if(A e. On, A, (/)) = suc x)))
9		0elon 2277	. . . . . 6 |- (/) e. On
10	9	elimel 1793	. . . . 5 |- if(A e. On, A, (/)) e. On
11	10	onuninsuc 2356	. . . 4 |- (if(A e. On, A, (/)) = U.if(A e. On, A, (/)) <-> -. E.x e. On if(A e. On, A, (/)) = suc x)
12	8, 11	dedth 1784	. . 3 |- (A e. On -> (A = U.A <-> -. E.x e. On A = suc x))
13		unon 2338	. . . . . 6 |- U.On = On
14	13	cleqcomi 1105	. . . . 5 |- On = U.On
15		onprc 2240	. . . . . . . 8 |- -. On e. V
16		visset 1350	. . . . . . . . . 10 |- x e. V
17	16	sucex 2303	. . . . . . . . 9 |- suc x e. V
18		eleq1 1149	. . . . . . . . 9 |- (On = suc x -> (On e. V <-> suc x e. V))
19	17, 18	mpbiri 169	. . . . . . . 8 |- (On = suc x -> On e. V)
20	15, 19	mto 93	. . . . . . 7 |- -. On = suc x
21	20	a1i 7	. . . . . 6 |- (x e. On -> -. On = suc x)
22	21	nrex 1270	. . . . 5 |- -. E.x e. On On = suc x
23	14, 22	2th 540	. . . 4 |- (On = U.On <-> -. E.x e. On On = suc x)
24		id 9	. . . . . 6 |- (A = On -> A = On)
25		unieq 1927	. . . . . 6 |- (A = On -> U.A = U.On)
26	24, 25	cleq12d 1115	. . . . 5 |- (A = On -> (A = U.A <-> On = U.On))
27		cleq1 1107	. . . . . . 7 |- (A = On -> (A = suc x <-> On = suc x))
28	27	birexdv 1220	. . . . . 6 |- (A = On -> (E.x e. On A = suc x <-> E.x e. On On = suc x))
29	28	negbid 463	. . . . 5 |- (A = On -> (-. E.x e. On A = suc x <-> -. E.x e. On On = suc x))
30	26, 29	bibi12d 477	. . . 4 |- (A = On -> ((A = U.A <-> -. E.x e. On A = suc x) <-> (On = U.On <-> -. E.x e. On On = suc x)))
31	23, 30	mpbiri 169	. . 3 |- (A = On -> (A = U.A <-> -. E.x e. On A = suc x))
32	12, 31	jaoi 275	. 2 |- ((A e. On \/ A = On) -> (A = U.A <-> -. E.x e. On A = suc x))
33	1, 32	sylbi 174	1 |- (Ord A -> (A = U.A <-> -. E.x e. On A = suc x))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   \/ wo 195   = wceq 1091   e. wcel 1092  E.wrex 1202  Vcvv 1348  (/)c0 1707  ifcif 1776  U.cuni 1919  Ord word 2198  Oncon0 2199  suc csuc 2201

This theorem is referenced by:  ordzsl 2366  dflim3 2368  nnsuc 2389  tfinds 2401

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-if 1777  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

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