Theorem sucprcreg 4743

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 3048	. 2 |- (-. A e. V -> suc A = A)
2		elirr 4742	. . . 4 |- -. A e. A
3		ax-17 1007	. . . . 5 |- (A e. A -> A.x A e. A)
4		eleq1 1577	. . . . 5 |- (x = A -> (x e. A <-> A e. A))
5	3, 4	ceqsalg 1871	. . . 4 |- (A e. V -> (A.x(x = A -> x e. A) <-> A e. A))
6	2, 5	mtbiri 722	. . 3 |- (A e. V -> -. A.x(x = A -> x e. A))
7		ssid 2132	. . . . . . . . 9 |- A (_ A
8		df-suc 2981	. . . . . . . . . . 11 |- suc A = (A u. {A})
9	8	eqeq1i 1525	. . . . . . . . . 10 |- (suc A = A <-> (A u. {A}) = A)
10		sseq1 2134	. . . . . . . . . 10 |- ((A u. {A}) = A -> ((A u. {A}) (_ A <-> A (_ A))
11	9, 10	sylbi 197	. . . . . . . . 9 |- (suc A = A -> ((A u. {A}) (_ A <-> A (_ A))
12	7, 11	mpbiri 192	. . . . . . . 8 |- (suc A = A -> (A u. {A}) (_ A)
13	12	sseld 2119	. . . . . . 7 |- (suc A = A -> (x e. (A u. {A}) -> x e. A))
14		elun 2225	. . . . . . 7 |- (x e. (A u. {A}) <-> (x e. A \/ x e. {A}))
15	13, 14	syl5ibr 205	. . . . . 6 |- (suc A = A -> ((x e. A \/ x e. {A}) -> x e. A))
16		olc 266	. . . . . 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 2479	. . . . 5 |- (x e. {A} <-> x = A)
19	17, 18	syl5ibr 205	. . . 4 |- (suc A = A -> (x = A -> x e. A))
20	19	19.21aiv 1324	. . 3 |- (suc A = A -> A.x(x = A -> x e. A))
21	6, 20	nsyl3 118	. 2 |- (suc A = A -> -. A e. V)
22	1, 21	impbii 155	1 |- (-. A e. V <-> suc A = A)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 144   \/ wo 220  A.wal 990   = wceq 992   e. wcel 994  Vcvv 1857   u. cun 2097   (_ wss 2099  {csn 2467  suc csuc 2977

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-sep 2777  ax-pow 2818  ax-reg 4736

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-v 1858  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-pw 2459  df-sn 2470  df-pr 2471  df-suc 2981

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