Theorem orduninsuc 3104

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 2980	. 2 |- (Ord A <-> (A e. On \/ A = On))
2		id 59	. . . . . 6 |- (A = if(A e. On, A, (/)) -> A = if(A e. On, A, (/)))
3		unieq 2500	. . . . . 6 |- (A = if(A e. On, A, (/)) -> U.A = U.if(A e. On, A, (/)))
4	2, 3	eqeq12d 1481	. . . . 5 |- (A = if(A e. On, A, (/)) -> (A = U.A <-> if(A e. On, A, (/)) = U.if(A e. On, A, (/))))
5		eqeq1 1473	. . . . . . 7 |- (A = if(A e. On, A, (/)) -> (A = suc x <-> if(A e. On, A, (/)) = suc x))
6	5	rexbidv 1656	. . . . . 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 609	. . . . 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 627	. . . 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 3012	. . . . . 6 |- (/) e. On
10	9	elimel 2384	. . . . 5 |- if(A e. On, A, (/)) e. On
11	10	onuninsuc 3098	. . . 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 2373	. . 3 |- (A e. On -> (A = U.A <-> -. E.x e. On A = suc x))
13		unon 3078	. . . . . 6 |- U.On = On
14	13	eqcomi 1471	. . . . 5 |- On = U.On
15		onprc 2979	. . . . . . . 8 |- -. On e. V
16		visset 1804	. . . . . . . . . 10 |- x e. V
17	16	sucex 3040	. . . . . . . . 9 |- suc x e. V
18		eleq1 1526	. . . . . . . . 9 |- (On = suc x -> (On e. V <-> suc x e. V))
19	17, 18	mpbiri 194	. . . . . . . 8 |- (On = suc x -> On e. V)
20	15, 19	mto 106	. . . . . . 7 |- -. On = suc x
21	20	a1i 8	. . . . . 6 |- (x e. On -> -. On = suc x)
22	21	nrex 1721	. . . . 5 |- -. E.x e. On On = suc x
23	14, 22	2th 716	. . . 4 |- (On = U.On <-> -. E.x e. On On = suc x)
24		id 59	. . . . . 6 |- (A = On -> A = On)
25		unieq 2500	. . . . . 6 |- (A = On -> U.A = U.On)
26	24, 25	eqeq12d 1481	. . . . 5 |- (A = On -> (A = U.A <-> On = U.On))
27		eqeq1 1473	. . . . . . 7 |- (A = On -> (A = suc x <-> On = suc x))
28	27	rexbidv 1656	. . . . . 6 |- (A = On -> (E.x e. On A = suc x <-> E.x e. On On = suc x))
29	28	negbid 609	. . . . 5 |- (A = On -> (-. E.x e. On A = suc x <-> -. E.x e. On On = suc x))
30	26, 29	bibi12d 627	. . . 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 194	. . 3 |- (A = On -> (A = U.A <-> -. E.x e. On A = suc x))
32	12, 31	jaoi 341	. 2 |- ((A e. On \/ A = On) -> (A = U.A <-> -. E.x e. On A = suc x))
33	1, 32	sylbi 199	1 |- (Ord A -> (A = U.A <-> -. E.x e. On A = suc x))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   = wceq 953   e. wcel 955  E.wrex 1638  Vcvv 1802  (/)c0 2270  ifcif 2351  U.cuni 2493  Ord word 2937  Oncon0 2938  suc csuc 2940

This theorem is referenced by:  ordunisuc2 3105  ordzsl 3106  dflim3 3108  nnsuc 3138  tfinds 3151

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-suc 2944

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