Theorem sucprcreg 3451

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 2297	. 2 |- (-. A e. V -> suc A = A)
2		eirr 3450	. . . 4 |- -. A e. A
3		ax-17 925	. . . . 5 |- (A e. A -> A.x A e. A)
4		eleq1 1149	. . . . 5 |- (x = A -> (x e. A <-> A e. A))
5	3, 4	ceqsalg 1362	. . . 4 |- (A e. V -> (A.x(x = A -> x e. A) <-> A e. A))
6	2, 5	mtbiri 539	. . 3 |- (A e. V -> -. A.x(x = A -> x e. A))
7		ssid 1519	. . . . . . . . 9 |- A (_ A
8		df-suc 2205	. . . . . . . . . . 11 |- suc A = (A u. {A})
9	8	cleq1i 1108	. . . . . . . . . 10 |- (suc A = A <-> (A u. {A}) = A)
10		sseq1 1521	. . . . . . . . . 10 |- ((A u. {A}) = A -> ((A u. {A}) (_ A <-> A (_ A))
11	9, 10	sylbi 174	. . . . . . . . 9 |- (suc A = A -> ((A u. {A}) (_ A <-> A (_ A))
12	7, 11	mpbiri 169	. . . . . . . 8 |- (suc A = A -> (A u. {A}) (_ A)
13	12	sseld 1506	. . . . . . 7 |- (suc A = A -> (x e. (A u. {A}) -> x e. A))
14		elun 1601	. . . . . . 7 |- (x e. (A u. {A}) <-> (x e. A \/ x e. {A}))
15	13, 14	syl5ibr 182	. . . . . 6 |- (suc A = A -> ((x e. A \/ x e. {A}) -> x e. A))
16		olc 224	. . . . . 6 |- (x e. {A} -> (x e. A \/ x e. {A}))
17	15, 16	syl5 22	. . . . 5 |- (suc A = A -> (x e. {A} -> x e. A))
18		elsn 1820	. . . . 5 |- (x e. {A} <-> x = A)
19	17, 18	syl5ibr 182	. . . 4 |- (suc A = A -> (x = A -> x e. A))
20	19	19.21aiv 943	. . 3 |- (suc A = A -> A.x(x = A -> x e. A))
21	6, 20	nsyl3 104	. 2 |- (suc A = A -> -. A e. V)
22	1, 21	impbi 139	1 |- (-. A e. V <-> suc A = A)

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   \/ wo 195  A.wal 672   = wceq 1091   e. wcel 1092  Vcvv 1348   u. cun 1485   (_ wss 1487  {csn 1808  suc csuc 2201

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077  ax-reg 1078

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-suc 2205

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