Theorem sucprcreg 4526

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 4390	. 2 |- ( -. A e. _V -> suc A = A )
2		elirr 4518	. . . 4 |- -. A e. A
3		nfv 1516	. . . . 5 |- F/ x A e. A
4		eleq1 2229	. . . . 5 |- ( x = A -> ( x e. A <-> A e. A ) )
5	3, 4	ceqsalg 2754	. . . 4 |- ( A e. _V -> ( A. x ( x = A -> x e. A ) <-> A e. A ) )
6	2, 5	mtbiri 665	. . 3 |- ( A e. _V -> -. A. x ( x = A -> x e. A ) )
7		velsn 3593	. . . . 5 |- ( x e. { A } <-> x = A )
8		olc 701	. . . . . 6 |- ( x e. { A } -> ( x e. A \/ x e. { A } ) )
9		elun 3263	. . . . . . 7 |- ( x e. ( A u. { A } ) <-> ( x e. A \/ x e. { A } ) )
10		ssid 3162	. . . . . . . . 9 |- A C_ A
11		df-suc 4349	. . . . . . . . . . 11 |- suc A = ( A u. { A } )
12	11	eqeq1i 2173	. . . . . . . . . 10 |- ( suc A = A <-> ( A u. { A } ) = A )
13		sseq1 3165	. . . . . . . . . 10 |- ( ( A u. { A } ) = A -> ( ( A u. { A } ) C_ A <-> A C_ A ) )
14	12, 13	sylbi 120	. . . . . . . . 9 |- ( suc A = A -> ( ( A u. { A } ) C_ A <-> A C_ A ) )
15	10, 14	mpbiri 167	. . . . . . . 8 |- ( suc A = A -> ( A u. { A } ) C_ A )
16	15	sseld 3141	. . . . . . 7 |- ( suc A = A -> ( x e. ( A u. { A } ) -> x e. A ) )
17	9, 16	syl5bir 152	. . . . . 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 152	. . . 4 |- ( suc A = A -> ( x = A -> x e. A ) )
20	19	alrimiv 1862	. . 3 |- ( suc A = A -> A. x ( x = A -> x e. A ) )
21	6, 20	nsyl3 616	. 2 |- ( suc A = A -> -. A e. _V )
22	1, 21	impbii 125	1 |- ( -. A e. _V <-> suc A = A )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104   \/ wo 698   A.wal 1341   = wceq 1343   e. wcel 2136   _Vcvv 2726   u. cun 3114   C_ wss 3116   {csn 3576   suc csuc 4343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-sn 3582  df-suc 4349

This theorem is referenced by: (None)