Theorem orduninsuc 3925

Description: An ordinal equal to its union is not a successor.

Assertion

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

Distinct variable group:   x,A

Proof of Theorem orduninsuc

Step	Hyp	Ref	Expression
1		ordeleqon 3866	. 2 |- (Ord A <-> (A e. On \/ A = On))
2		id 73	. . . . . 6 |- (A = if(A e. On, A, (/)) -> A = if(A e. On, A, (/)))
3		unieq 3185	. . . . . 6 |- (A = if(A e. On, A, (/)) -> U.A = U.if(A e. On, A, (/)))
4	2, 3	eqeq12d 1899	. . . . 5 |- (A = if(A e. On, A, (/)) -> (A = U.A <-> if(A e. On, A, (/)) = U.if(A e. On, A, (/))))
5		eqeq1 1890	. . . . . . 7 |- (A = if(A e. On, A, (/)) -> (A = suc x <-> if(A e. On, A, (/)) = suc x))
6	5	rexbidv 2124	. . . . . 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	notbid 673	. . . . 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 691	. . . 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 3716	. . . . . 6 |- (/) e. On
10	9	elimel 3025	. . . . 5 |- if(A e. On, A, (/)) e. On
11	10	onuninsuci 3921	. . . 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 3011	. . 3 |- (A e. On -> (A = U.A <-> -. E.x e. On A = suc x))
13		unon 3910	. . . . . 6 |- U.On = On
14	13	eqcomi 1888	. . . . 5 |- On = U.On
15		onprc 3865	. . . . . . . 8 |- -. On e. _V
16		visset 2295	. . . . . . . . . 10 |- x e. _V
17	16	sucex 3892	. . . . . . . . 9 |- suc x e. _V
18		eleq1 1957	. . . . . . . . 9 |- (On = suc x -> (On e. _V <-> suc x e. _V))
19	17, 18	mpbiri 211	. . . . . . . 8 |- (On = suc x -> On e. _V)
20	15, 19	mto 121	. . . . . . 7 |- -. On = suc x
21	20	a1i 8	. . . . . 6 |- (x e. On -> -. On = suc x)
22	21	nrex 2192	. . . . 5 |- -. E.x e. On On = suc x
23	14, 22	2th 786	. . . 4 |- (On = U.On <-> -. E.x e. On On = suc x)
24		id 73	. . . . . 6 |- (A = On -> A = On)
25		unieq 3185	. . . . . 6 |- (A = On -> U.A = U.On)
26	24, 25	eqeq12d 1899	. . . . 5 |- (A = On -> (A = U.A <-> On = U.On))
27		eqeq1 1890	. . . . . . 7 |- (A = On -> (A = suc x <-> On = suc x))
28	27	rexbidv 2124	. . . . . 6 |- (A = On -> (E.x e. On A = suc x <-> E.x e. On On = suc x))
29	28	notbid 673	. . . . 5 |- (A = On -> (-. E.x e. On A = suc x <-> -. E.x e. On On = suc x))
30	26, 29	bibi12d 691	. . . 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 211	. . 3 |- (A = On -> (A = U.A <-> -. E.x e. On A = suc x))
32	12, 31	jaoi 368	. 2 |- ((A e. On \/ A = On) -> (A = U.A <-> -. E.x e. On A = suc x))
33	1, 32	sylbi 216	1 |- (Ord A -> (A = U.A <-> -. E.x e. On A = suc x))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   = wceq 1298   e. wcel 1300  E.wrex 2106  _Vcvv 2292  (/)c0 2875  ifcif 2982  U.cuni 3177  Ord word 3656  Oncon0 3657  suc csuc 3659

This theorem is referenced by:  ordunisuc2 3926  ordzsl 3927  dflim3 3930  dflim3OLD 3931  tfindsOLD 3943  nnsuc 3969

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-if 2983  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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