Theorem sucprcreg 4600

Description: A class is equal to its successor iff it is a proper class (assuming the Axiom of Regularity).

Assertion

Ref	Expression
sucprcreg	|- (-. A e. V <-> suc A = A)

Proof of Theorem sucprcreg

Step	Hyp	Ref	Expression
1		sucprc 3044	. 2 |- (-. A e. V -> suc A = A)
2		elirr 4599	. . . 4 |- -. A e. A
3		ax-17 971	. . . . 5 |- (A e. A -> A.x A e. A)
4		eleq1 1534	. . . . 5 |- (x = A -> (x e. A <-> A e. A))
5	3, 4	ceqsalg 1825	. . . 4 |- (A e. V -> (A.x(x = A -> x e. A) <-> A e. A))
6	2, 5	mtbiri 717	. . 3 |- (A e. V -> -. A.x(x = A -> x e. A))
7		ssid 2080	. . . . . . . . 9 |- A (_ A
8		df-suc 2954	. . . . . . . . . . 11 |- suc A = (A u. {A})
9	8	eqeq1i 1482	. . . . . . . . . 10 |- (suc A = A <-> (A u. {A}) = A)
10		sseq1 2082	. . . . . . . . . 10 |- ((A u. {A}) = A -> ((A u. {A}) (_ A <-> A (_ A))
11	9, 10	sylbi 199	. . . . . . . . 9 |- (suc A = A -> ((A u. {A}) (_ A <-> A (_ A))
12	7, 11	mpbiri 194	. . . . . . . 8 |- (suc A = A -> (A u. {A}) (_ A)
13	12	sseld 2067	. . . . . . 7 |- (suc A = A -> (x e. (A u. {A}) -> x e. A))
14		elun 2173	. . . . . . 7 |- (x e. (A u. {A}) <-> (x e. A \/ x e. {A}))
15	13, 14	syl5ibr 207	. . . . . 6 |- (suc A = A -> ((x e. A \/ x e. {A}) -> x e. A))
16		olc 268	. . . . . 6 |- (x e. {A} -> (x e. A \/ x e. {A}))
17	15, 16	syl5 21	. . . . 5 |- (suc A = A -> (x e. {A} -> x e. A))
18		elsn 2421	. . . . 5 |- (x e. {A} <-> x = A)
19	17, 18	syl5ibr 207	. . . 4 |- (suc A = A -> (x = A -> x e. A))
20	19	19.21aiv 1286	. . 3 |- (suc A = A -> A.x(x = A -> x e. A))
21	6, 20	nsyl3 119	. 2 |- (suc A = A -> -. A e. V)
22	1, 21	impbi 157	1 |- (-. A e. V <-> suc A = A)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222  A.wal 954   = wceq 956   e. wcel 958  Vcvv 1811   u. cun 2045   (_ wss 2047  {csn 2409  suc csuc 2950

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-reg 4593

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-suc 2954

Copyright terms: Public domain