Theorem sucprcreg 4393

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 4263	. 2 |- ( -. A e. _V -> suc A = A )
2		elirr 4385	. . . 4 |- -. A e. A
3		nfv 1473	. . . . 5 |- F/ x A e. A
4		eleq1 2157	. . . . 5 |- ( x = A -> ( x e. A <-> A e. A ) )
5	3, 4	ceqsalg 2661	. . . 4 |- ( A e. _V -> ( A. x ( x = A -> x e. A ) <-> A e. A ) )
6	2, 5	mtbiri 638	. . 3 |- ( A e. _V -> -. A. x ( x = A -> x e. A ) )
7		velsn 3483	. . . . 5 |- ( x e. { A } <-> x = A )
8		olc 670	. . . . . 6 |- ( x e. { A } -> ( x e. A \/ x e. { A } ) )
9		elun 3156	. . . . . . 7 |- ( x e. ( A u. { A } ) <-> ( x e. A \/ x e. { A } ) )
10		ssid 3059	. . . . . . . . 9 |- A C_ A
11		df-suc 4222	. . . . . . . . . . 11 |- suc A = ( A u. { A } )
12	11	eqeq1i 2102	. . . . . . . . . 10 |- ( suc A = A <-> ( A u. { A } ) = A )
13		sseq1 3062	. . . . . . . . . 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 3038	. . . . . . 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 1809	. . 3 |- ( suc A = A -> A. x ( x = A -> x e. A ) )
21	6, 20	nsyl3 594	. 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 667   A.wal 1294   = wceq 1296   e. wcel 1445   _Vcvv 2633   u. cun 3011   C_ wss 3013   {csn 3466   suc csuc 4216

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-setind 4381

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-v 2635  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-sn 3472  df-suc 4222

This theorem is referenced by: (None)