Theorem sucprcreg 4694

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 4555	. 2 |- ( -. A e. _V -> suc A = A )
2		elirr 4686	. . . 4 |- -. A e. A
3		nfv 1581	. . . . 5 |- F/ x A e. A
4		eleq1 2301	. . . . 5 |- ( x = A -> ( x e. A <-> A e. A ) )
5	3, 4	ceqsalg 2850	. . . 4 |- ( A e. _V -> ( A. x ( x = A -> x e. A ) <-> A e. A ) )
6	2, 5	mtbiri 686	. . 3 |- ( A e. _V -> -. A. x ( x = A -> x e. A ) )
7		velsn 3725	. . . . 5 |- ( x e. { A } <-> x = A )
8		olc 723	. . . . . 6 |- ( x e. { A } -> ( x e. A \/ x e. { A } ) )
9		elun 3370	. . . . . . 7 |- ( x e. ( A u. { A } ) <-> ( x e. A \/ x e. { A } ) )
10		ssid 3268	. . . . . . . . 9 |- A C_ A
11		df-suc 4514	. . . . . . . . . . 11 |- suc A = ( A u. { A } )
12	11	eqeq1i 2246	. . . . . . . . . 10 |- ( suc A = A <-> ( A u. { A } ) = A )
13		sseq1 3271	. . . . . . . . . 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 3247	. . . . . . 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 1927	. . 3 |- ( suc A = A -> A. x ( x = A -> x e. A ) )
21	6, 20	nsyl3 635	. 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 720   A.wal 1400   = wceq 1402   e. wcel 2209   _Vcvv 2821   u. cun 3218   C_ wss 3220   {csn 3708   suc csuc 4508

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-setind 4682

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-sn 3714  df-suc 4514

This theorem is referenced by: (None)