Theorem sucprcreg 4227

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 4115	. 2 |- (-. A e. _V -> suc A = A)
2		elirr 4224	. . . 4 |- -. A e. A
3		nfv 1418	. . . . 5 |- F/x A e. A
4		eleq1 2097	. . . . 5 |- (x = A -> (x e. A <-> A e. A))
5	3, 4	ceqsalg 2576	. . . 4 |- (A e. _V -> (A.x(x = A -> x e. A) <-> A e. A))
6	2, 5	mtbiri 599	. . 3 |- (A e. _V -> -. A.x(x = A -> x e. A))
7		elsn 3382	. . . . 5 |- (x e. {A } <-> x = A)
8		olc 631	. . . . . 6 |- (x e. {A } -> (x e. A \/ x e. {A }))
9		elun 3078	. . . . . . 7 |- (x e. (A u. {A }) <-> (x e. A \/ x e. {A }))
10		ssid 2958	. . . . . . . . 9 |- A C_ A
11		df-suc 4074	. . . . . . . . . . 11 |- suc A = (A u. {A })
12	11	eqeq1i 2044	. . . . . . . . . 10 |- ( suc A = A <-> (A u. {A }) = A)
13		sseq1 2960	. . . . . . . . . 10 |- ((A u. {A }) = A -> ((A u. {A }) C_ A <-> A C_ A))
14	12, 13	sylbi 114	. . . . . . . . 9 |- ( suc A = A -> ((A u. {A }) C_ A <-> A C_ A))
15	10, 14	mpbiri 157	. . . . . . . 8 |- ( suc A = A -> (A u. {A }) C_ A)
16	15	sseld 2938	. . . . . . 7 |- ( suc A = A -> (x e. (A u. {A }) -> x e. A))
17	9, 16	syl5bir 142	. . . . . 6 |- ( suc A = A -> ((x e. A \/ x e. {A }) -> x e. A))
18	8, 17	syl5 28	. . . . 5 |- ( suc A = A -> (x e. {A } -> x e. A))
19	7, 18	syl5bir 142	. . . 4 |- ( suc A = A -> (x = A -> x e. A))
20	19	alrimiv 1751	. . 3 |- ( suc A = A -> A.x(x = A -> x e. A))
21	6, 20	nsyl3 556	. 2 |- ( suc A = A -> -. A e. _V)
22	1, 21	impbii 117	1 |- (-. A e. _V <-> suc A = A)

Syntax hints:  -. wn 3   -> wi 4   <-> wb 98   \/ wo 628  A.wal 1240   = wceq 1242   e. wcel 1390   _Vcvv 2551   u. cun 2909   C_ wss 2911   {csn 3367   suc csuc 4068

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-setind 4220

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-sn 3373  df-suc 4074

This theorem is referenced by: (None)