Theorem sucprcreg 4598

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 4460	. 2 |- ( -. A e. _V -> suc A = A )
2		elirr 4590	. . . 4 |- -. A e. A
3		nfv 1551	. . . . 5 |- F/ x A e. A
4		eleq1 2268	. . . . 5 |- ( x = A -> ( x e. A <-> A e. A ) )
5	3, 4	ceqsalg 2800	. . . 4 |- ( A e. _V -> ( A. x ( x = A -> x e. A ) <-> A e. A ) )
6	2, 5	mtbiri 677	. . 3 |- ( A e. _V -> -. A. x ( x = A -> x e. A ) )
7		velsn 3650	. . . . 5 |- ( x e. { A } <-> x = A )
8		olc 713	. . . . . 6 |- ( x e. { A } -> ( x e. A \/ x e. { A } ) )
9		elun 3314	. . . . . . 7 |- ( x e. ( A u. { A } ) <-> ( x e. A \/ x e. { A } ) )
10		ssid 3213	. . . . . . . . 9 |- A C_ A
11		df-suc 4419	. . . . . . . . . . 11 |- suc A = ( A u. { A } )
12	11	eqeq1i 2213	. . . . . . . . . 10 |- ( suc A = A <-> ( A u. { A } ) = A )
13		sseq1 3216	. . . . . . . . . 10 |- ( ( A u. { A } ) = A -> ( ( A u. { A } ) C_ A <-> A C_ A ) )
14	12, 13	sylbi 121	. . . . . . . . 9 |- ( suc A = A -> ( ( A u. { A } ) C_ A <-> A C_ A ) )
15	10, 14	mpbiri 168	. . . . . . . 8 |- ( suc A = A -> ( A u. { A } ) C_ A )
16	15	sseld 3192	. . . . . . 7 |- ( suc A = A -> ( x e. ( A u. { A } ) -> x e. A ) )
17	9, 16	biimtrrid 153	. . . . . 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	biimtrrid 153	. . . 4 |- ( suc A = A -> ( x = A -> x e. A ) )
20	19	alrimiv 1897	. . 3 |- ( suc A = A -> A. x ( x = A -> x e. A ) )
21	6, 20	nsyl3 627	. 2 |- ( suc A = A -> -. A e. _V )
22	1, 21	impbii 126	1 |- ( -. A e. _V <-> suc A = A )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   \/ wo 710   A.wal 1371   = wceq 1373   e. wcel 2176   _Vcvv 2772   u. cun 3164   C_ wss 3166   {csn 3633   suc csuc 4413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187  ax-setind 4586

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-sn 3639  df-suc 4419

This theorem is referenced by: (None)