Theorem orduninsuc 3114

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 2990	. 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 2510	. . . . . 6 |- (A = if(A e. On, A, (/)) -> U.A = U.if(A e. On, A, (/)))
4	2, 3	eqeq12d 1489	. . . . 5 |- (A = if(A e. On, A, (/)) -> (A = U.A <-> if(A e. On, A, (/)) = U.if(A e. On, A, (/))))
5		eqeq1 1481	. . . . . . 7 |- (A = if(A e. On, A, (/)) -> (A = suc x <-> if(A e. On, A, (/)) = suc x))
6	5	rexbidv 1664	. . . . . 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 611	. . . . 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 629	. . . 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 3022	. . . . . 6 |- (/) e. On
10	9	elimel 2394	. . . . 5 |- if(A e. On, A, (/)) e. On
11	10	onuninsuc 3108	. . . 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 2383	. . 3 |- (A e. On -> (A = U.A <-> -. E.x e. On A = suc x))
13		unon 3088	. . . . . 6 |- U.On = On
14	13	eqcomi 1479	. . . . 5 |- On = U.On
15		onprc 2989	. . . . . . . 8 |- -. On e. V
16		visset 1813	. . . . . . . . . 10 |- x e. V
17	16	sucex 3050	. . . . . . . . 9 |- suc x e. V
18		eleq1 1534	. . . . . . . . 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 1729	. . . . 5 |- -. E.x e. On On = suc x
23	14, 22	2th 718	. . . 4 |- (On = U.On <-> -. E.x e. On On = suc x)
24		id 59	. . . . . 6 |- (A = On -> A = On)
25		unieq 2510	. . . . . 6 |- (A = On -> U.A = U.On)
26	24, 25	eqeq12d 1489	. . . . 5 |- (A = On -> (A = U.A <-> On = U.On))
27		eqeq1 1481	. . . . . . 7 |- (A = On -> (A = suc x <-> On = suc x))
28	27	rexbidv 1664	. . . . . 6 |- (A = On -> (E.x e. On A = suc x <-> E.x e. On On = suc x))
29	28	negbid 611	. . . . 5 |- (A = On -> (-. E.x e. On A = suc x <-> -. E.x e. On On = suc x))
30	26, 29	bibi12d 629	. . . 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 956   e. wcel 958  E.wrex 1646  Vcvv 1811  (/)c0 2280  ifcif 2361  U.cuni 2503  Ord word 2947  Oncon0 2948  suc csuc 2950

This theorem is referenced by:  ordunisuc2 3115  ordzsl 3116  dflim3 3118  nnsuc 3148  tfinds 3161

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-suc 2954

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