Theorem orduninsuc 3110

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 2986	. 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 2506	. . . . . 6 |- (A = if(A e. On, A, (/)) -> U.A = U.if(A e. On, A, (/)))
4	2, 3	eqeq12d 1487	. . . . 5 |- (A = if(A e. On, A, (/)) -> (A = U.A <-> if(A e. On, A, (/)) = U.if(A e. On, A, (/))))
5		eqeq1 1479	. . . . . . 7 |- (A = if(A e. On, A, (/)) -> (A = suc x <-> if(A e. On, A, (/)) = suc x))
6	5	rexbidv 1662	. . . . . 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 610	. . . . 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 628	. . . 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 3018	. . . . . 6 |- (/) e. On
10	9	elimel 2391	. . . . 5 |- if(A e. On, A, (/)) e. On
11	10	onuninsuc 3104	. . . 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 2380	. . 3 |- (A e. On -> (A = U.A <-> -. E.x e. On A = suc x))
13		unon 3084	. . . . . 6 |- U.On = On
14	13	eqcomi 1477	. . . . 5 |- On = U.On
15		onprc 2985	. . . . . . . 8 |- -. On e. V
16		visset 1810	. . . . . . . . . 10 |- x e. V
17	16	sucex 3046	. . . . . . . . 9 |- suc x e. V
18		eleq1 1532	. . . . . . . . 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 1727	. . . . 5 |- -. E.x e. On On = suc x
23	14, 22	2th 717	. . . 4 |- (On = U.On <-> -. E.x e. On On = suc x)
24		id 59	. . . . . 6 |- (A = On -> A = On)
25		unieq 2506	. . . . . 6 |- (A = On -> U.A = U.On)
26	24, 25	eqeq12d 1487	. . . . 5 |- (A = On -> (A = U.A <-> On = U.On))
27		eqeq1 1479	. . . . . . 7 |- (A = On -> (A = suc x <-> On = suc x))
28	27	rexbidv 1662	. . . . . 6 |- (A = On -> (E.x e. On A = suc x <-> E.x e. On On = suc x))
29	28	negbid 610	. . . . 5 |- (A = On -> (-. E.x e. On A = suc x <-> -. E.x e. On On = suc x))
30	26, 29	bibi12d 628	. . . 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 955   e. wcel 957  E.wrex 1644  Vcvv 1808  (/)c0 2277  ifcif 2358  U.cuni 2499  Ord word 2943  Oncon0 2944  suc csuc 2946

This theorem is referenced by:  ordunisuc2 3111  ordzsl 3112  dflim3 3114  nnsuc 3144  tfinds 3157

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-if 2359  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-suc 2950

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