Theorem sucprcreg 7979

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 4896	. 2 |- ( -. A e. _V -> suc A = A )
2		elirr 7978	. . . 4 |- -. A e. A
3		df-suc 4827	. . . . . . . 8 |- suc A = ( A u. { A } )
4	3	eqeq1i 2409	. . . . . . 7 |- ( suc A = A <-> ( A u. { A } ) = A )
5		ssequn2 3615	. . . . . . 7 |- ( { A } C_ A <-> ( A u. { A } ) = A )
6	4, 5	bitr4i 252	. . . . . 6 |- ( suc A = A <-> { A } C_ A )
7	6	biimpi 194	. . . . 5 |- ( suc A = A -> { A } C_ A )
8		snidg 3997	. . . . 5 |- ( A e. _V -> A e. { A } )
9		ssel2 3436	. . . . 5 |- ( ( { A } C_ A /\ A e. { A } ) -> A e. A )
10	7, 8, 9	syl2an 475	. . . 4 |- ( ( suc A = A /\ A e. _V ) -> A e. A )
11	2, 10	mto 176	. . 3 |- -. ( suc A = A /\ A e. _V )
12	11	imnani 421	. 2 |- ( suc A = A -> -. A e. _V )
13	1, 12	impbii 188	1 |- ( -. A e. _V <-> suc A = A )

Syntax hints:   -. wn 3   <-> wb 184   /\ wa 367   = wceq 1405   e. wcel 1842   _Vcvv 3058   u. cun 3411   C_ wss 3413   {csn 3971   suc csuc 4823

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629  ax-reg 7972

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-sn 3972  df-pr 3974  df-suc 4827

This theorem is referenced by: (None)