Theorem onuninsuci 7040

Description: A limit ordinal is not a successor ordinal. (Contributed by NM, 18-Feb-2004.)

Hypothesis

Ref	Expression
onssi.1	|- A e. On

Assertion

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

Distinct variable group:   x, A

Proof of Theorem onuninsuci

Step	Hyp	Ref	Expression
1		onssi.1	. . . . . . 7 |- A e. On
2	1	onirri 5834	. . . . . 6 |- -. A e. A
3		id 22	. . . . . . . 8 |- ( A = U. A -> A = U. A )
4		df-suc 5729	. . . . . . . . . . . 12 |- suc x = ( x u. { x } )
5	4	eqeq2i 2634	. . . . . . . . . . 11 |- ( A = suc x <-> A = ( x u. { x } ) )
6		unieq 4444	. . . . . . . . . . 11 |- ( A = ( x u. { x } ) -> U. A = U. ( x u. { x } ) )
7	5, 6	sylbi 207	. . . . . . . . . 10 |- ( A = suc x -> U. A = U. ( x u. { x } ) )
8		uniun 4456	. . . . . . . . . . 11 |- U. ( x u. { x } ) = ( U. x u. U. { x } )
9		vex 3203	. . . . . . . . . . . . 13 |- x e. _V
10	9	unisn 4451	. . . . . . . . . . . 12 |- U. { x } = x
11	10	uneq2i 3764	. . . . . . . . . . 11 |- ( U. x u. U. { x } ) = ( U. x u. x )
12	8, 11	eqtri 2644	. . . . . . . . . 10 |- U. ( x u. { x } ) = ( U. x u. x )
13	7, 12	syl6eq 2672	. . . . . . . . 9 |- ( A = suc x -> U. A = ( U. x u. x ) )
14		tron 5746	. . . . . . . . . . . 12 |- Tr On
15		eleq1 2689	. . . . . . . . . . . . 13 |- ( A = suc x -> ( A e. On <-> suc x e. On ) )
16	1, 15	mpbii 223	. . . . . . . . . . . 12 |- ( A = suc x -> suc x e. On )
17		trsuc 5810	. . . . . . . . . . . 12 |- ( ( Tr On /\ suc x e. On ) -> x e. On )
18	14, 16, 17	sylancr 695	. . . . . . . . . . 11 |- ( A = suc x -> x e. On )
19		eloni 5733	. . . . . . . . . . . . 13 |- ( x e. On -> Ord x )
20		ordtr 5737	. . . . . . . . . . . . 13 |- ( Ord x -> Tr x )
21	19, 20	syl 17	. . . . . . . . . . . 12 |- ( x e. On -> Tr x )
22		df-tr 4753	. . . . . . . . . . . 12 |- ( Tr x <-> U. x C_ x )
23	21, 22	sylib 208	. . . . . . . . . . 11 |- ( x e. On -> U. x C_ x )
24	18, 23	syl 17	. . . . . . . . . 10 |- ( A = suc x -> U. x C_ x )
25		ssequn1 3783	. . . . . . . . . 10 |- ( U. x C_ x <-> ( U. x u. x ) = x )
26	24, 25	sylib 208	. . . . . . . . 9 |- ( A = suc x -> ( U. x u. x ) = x )
27	13, 26	eqtrd 2656	. . . . . . . 8 |- ( A = suc x -> U. A = x )
28	3, 27	sylan9eqr 2678	. . . . . . 7 |- ( ( A = suc x /\ A = U. A ) -> A = x )
29	9	sucid 5804	. . . . . . . . 9 |- x e. suc x
30		eleq2 2690	. . . . . . . . 9 |- ( A = suc x -> ( x e. A <-> x e. suc x ) )
31	29, 30	mpbiri 248	. . . . . . . 8 |- ( A = suc x -> x e. A )
32	31	adantr 481	. . . . . . 7 |- ( ( A = suc x /\ A = U. A ) -> x e. A )
33	28, 32	eqeltrd 2701	. . . . . 6 |- ( ( A = suc x /\ A = U. A ) -> A e. A )
34	2, 33	mto 188	. . . . 5 |- -. ( A = suc x /\ A = U. A )
35	34	imnani 439	. . . 4 |- ( A = suc x -> -. A = U. A )
36	35	rexlimivw 3029	. . 3 |- ( E. x e. On A = suc x -> -. A = U. A )
37		onuni 6993	. . . . 5 |- ( A e. On -> U. A e. On )
38	1, 37	ax-mp 5	. . . 4 |- U. A e. On
39	1	onuniorsuci 7039	. . . . 5 |- ( A = U. A \/ A = suc U. A )
40	39	ori 390	. . . 4 |- ( -. A = U. A -> A = suc U. A )
41		suceq 5790	. . . . . 6 |- ( x = U. A -> suc x = suc U. A )
42	41	eqeq2d 2632	. . . . 5 |- ( x = U. A -> ( A = suc x <-> A = suc U. A ) )
43	42	rspcev 3309	. . . 4 |- ( ( U. A e. On /\ A = suc U. A ) -> E. x e. On A = suc x )
44	38, 40, 43	sylancr 695	. . 3 |- ( -. A = U. A -> E. x e. On A = suc x )
45	36, 44	impbii 199	. 2 |- ( E. x e. On A = suc x <-> -. A = U. A )
46	45	con2bii 347	1 |- ( A = U. A <-> -. E. x e. On A = suc x )

Syntax hints:   -. wn 3   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   u. cun 3572   C_ wss 3574   {csn 4177   U.cuni 4436   Tr wtr 4752   Ord word 5722   Oncon0 5723   suc csuc 5725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-suc 5729

This theorem is referenced by:  orduninsuc  7043