Theorem sucprcreg 8112

Description: A class is equal to its successor iff it is a proper class (assuming the Axiom of Regularity). (Contributed by NM, 9-Jul-2004.) (Proof shortened by BJ, 16-Apr-2019.)

Assertion

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

Proof of Theorem sucprcreg

Step	Hyp	Ref	Expression
1		sucprc 5509	. 2 |- ( -. A e. _V -> suc A = A )
2		elirr 8111	. . . 4 |- -. A e. A
3		df-suc 5440	. . . . . . . 8 |- suc A = ( A u. { A } )
4	3	eqeq1i 2427	. . . . . . 7 |- ( suc A = A <-> ( A u. { A } ) = A )
5		ssequn2 3636	. . . . . . 7 |- ( { A } C_ A <-> ( A u. { A } ) = A )
6	4, 5	bitr4i 255	. . . . . 6 |- ( suc A = A <-> { A } C_ A )
7	6	biimpi 197	. . . . 5 |- ( suc A = A -> { A } C_ A )
8		snidg 4019	. . . . 5 |- ( A e. _V -> A e. { A } )
9		ssel2 3456	. . . . 5 |- ( ( { A } C_ A /\ A e. { A } ) -> A e. A )
10	7, 8, 9	syl2an 479	. . . 4 |- ( ( suc A = A /\ A e. _V ) -> A e. A )
11	2, 10	mto 179	. . 3 |- -. ( suc A = A /\ A e. _V )
12	11	imnani 424	. 2 |- ( suc A = A -> -. A e. _V )
13	1, 12	impbii 190	1 |- ( -. A e. _V <-> suc A = A )

Syntax hints:   -. wn 3   <-> wb 187   /\ wa 370   = wceq 1437   e. wcel 1867   _Vcvv 3078   u. cun 3431   C_ wss 3433   {csn 3993   suc csuc 5436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4540  ax-nul 4548  ax-pr 4653  ax-reg 8105

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-sn 3994  df-pr 3996  df-suc 5440

This theorem is referenced by: (None)