Theorem sucprcreg 4292

Description: A class is equal to its successor iff it is a proper class (assuming the Axiom of Set Induction). (Contributed by NM, 9-Jul-2004.)

Assertion

Ref	Expression
sucprcreg	|- ( -. A e. _V <-> suc A = A )

Proof of Theorem sucprcreg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sucprc 4167	. 2 |- ( -. A e. _V -> suc A = A )
2		elirr 4284	. . . 4 |- -. A e. A
3		nfv 1461	. . . . 5 |- F/ x A e. A
4		eleq1 2141	. . . . 5 |- ( x = A -> ( x e. A <-> A e. A ) )
5	3, 4	ceqsalg 2627	. . . 4 |- ( A e. _V -> ( A. x ( x = A -> x e. A ) <-> A e. A ) )
6	2, 5	mtbiri 632	. . 3 |- ( A e. _V -> -. A. x ( x = A -> x e. A ) )
7		velsn 3415	. . . . 5 |- ( x e. { A } <-> x = A )
8		olc 664	. . . . . 6 |- ( x e. { A } -> ( x e. A \/ x e. { A } ) )
9		elun 3113	. . . . . . 7 |- ( x e. ( A u. { A } ) <-> ( x e. A \/ x e. { A } ) )
10		ssid 3018	. . . . . . . . 9 |- A C_ A
11		df-suc 4126	. . . . . . . . . . 11 |- suc A = ( A u. { A } )
12	11	eqeq1i 2088	. . . . . . . . . 10 |- ( suc A = A <-> ( A u. { A } ) = A )
13		sseq1 3020	. . . . . . . . . 10 |- ( ( A u. { A } ) = A -> ( ( A u. { A } ) C_ A <-> A C_ A ) )
14	12, 13	sylbi 119	. . . . . . . . 9 |- ( suc A = A -> ( ( A u. { A } ) C_ A <-> A C_ A ) )
15	10, 14	mpbiri 166	. . . . . . . 8 |- ( suc A = A -> ( A u. { A } ) C_ A )
16	15	sseld 2998	. . . . . . 7 |- ( suc A = A -> ( x e. ( A u. { A } ) -> x e. A ) )
17	9, 16	syl5bir 151	. . . . . 6 |- ( suc A = A -> ( ( x e. A \/ x e. { A } ) -> x e. A ) )
18	8, 17	syl5 32	. . . . 5 |- ( suc A = A -> ( x e. { A } -> x e. A ) )
19	7, 18	syl5bir 151	. . . 4 |- ( suc A = A -> ( x = A -> x e. A ) )
20	19	alrimiv 1795	. . 3 |- ( suc A = A -> A. x ( x = A -> x e. A ) )
21	6, 20	nsyl3 588	. 2 |- ( suc A = A -> -. A e. _V )
22	1, 21	impbii 124	1 |- ( -. A e. _V <-> suc A = A )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 103   \/ wo 661   A.wal 1282   = wceq 1284   e. wcel 1433   _Vcvv 2601   u. cun 2971   C_ wss 2973   {csn 3398   suc csuc 4120

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-sn 3404  df-suc 4126

This theorem is referenced by: (None)