Theorem orduninsuc 7043

Description: An ordinal equal to its union is not a successor. (Contributed by NM, 18-Feb-2004.)

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 6988	. 2 |- ( Ord A <-> ( A e. On \/ A = On ) )
2		id 22	. . . . . 6 |- ( A = if ( A e. On , A , (/) ) -> A = if ( A e. On , A , (/) ) )
3		unieq 4444	. . . . . 6 |- ( A = if ( A e. On , A , (/) ) -> U. A = U. if ( A e. On , A , (/) ) )
4	2, 3	eqeq12d 2637	. . . . 5 |- ( A = if ( A e. On , A , (/) ) -> ( A = U. A <-> if ( A e. On , A , (/) ) = U. if ( A e. On , A , (/) ) ) )
5		eqeq1 2626	. . . . . . 7 |- ( A = if ( A e. On , A , (/) ) -> ( A = suc x <-> if ( A e. On , A , (/) ) = suc x ) )
6	5	rexbidv 3052	. . . . . 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 308	. . . . 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 335	. . . 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 5778	. . . . . 6 |- (/) e. On
10	9	elimel 4150	. . . . 5 |- if ( A e. On , A , (/) ) e. On
11	10	onuninsuci 7040	. . . 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 4139	. . 3 |- ( A e. On -> ( A = U. A <-> -. E. x e. On A = suc x ) )
13		unon 7031	. . . . . 6 |- U. On = On
14	13	eqcomi 2631	. . . . 5 |- On = U. On
15		onprc 6984	. . . . . . . 8 |- -. On e. _V
16		vex 3203	. . . . . . . . . 10 |- x e. _V
17	16	sucex 7011	. . . . . . . . 9 |- suc x e. _V
18		eleq1 2689	. . . . . . . . 9 |- ( On = suc x -> ( On e. _V <-> suc x e. _V ) )
19	17, 18	mpbiri 248	. . . . . . . 8 |- ( On = suc x -> On e. _V )
20	15, 19	mto 188	. . . . . . 7 |- -. On = suc x
21	20	a1i 11	. . . . . 6 |- ( x e. On -> -. On = suc x )
22	21	nrex 3000	. . . . 5 |- -. E. x e. On On = suc x
23	14, 22	2th 254	. . . 4 |- ( On = U. On <-> -. E. x e. On On = suc x )
24		id 22	. . . . . 6 |- ( A = On -> A = On )
25		unieq 4444	. . . . . 6 |- ( A = On -> U. A = U. On )
26	24, 25	eqeq12d 2637	. . . . 5 |- ( A = On -> ( A = U. A <-> On = U. On ) )
27		eqeq1 2626	. . . . . . 7 |- ( A = On -> ( A = suc x <-> On = suc x ) )
28	27	rexbidv 3052	. . . . . 6 |- ( A = On -> ( E. x e. On A = suc x <-> E. x e. On On = suc x ) )
29	28	notbid 308	. . . . 5 |- ( A = On -> ( -. E. x e. On A = suc x <-> -. E. x e. On On = suc x ) )
30	26, 29	bibi12d 335	. . . 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 248	. . 3 |- ( A = On -> ( A = U. A <-> -. E. x e. On A = suc x ) )
32	12, 31	jaoi 394	. 2 |- ( ( A e. On \/ A = On ) -> ( A = U. A <-> -. E. x e. On A = suc x ) )
33	1, 32	sylbi 207	1 |- ( Ord A -> ( A = U. A <-> -. E. x e. On A = suc x ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   (/)c0 3915   ifcif 4086   U.cuni 4436   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-pw 4160  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:  ordunisuc2  7044  ordzsl  7045  dflim3  7047  nnsuc  7082