Theorem sucprcreg 7929

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 4905	. 2 |- ( -. A e. _V -> suc A = A )
2		elirr 7928	. . . 4 |- -. A e. A
3		df-suc 4836	. . . . . . . 8 |- suc A = ( A u. { A } )
4	3	eqeq1i 2461	. . . . . . 7 |- ( suc A = A <-> ( A u. { A } ) = A )
5		ssequn2 3640	. . . . . . 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 4014	. . . . 5 |- ( A e. _V -> A e. { A } )
9		ssel2 3462	. . . . 5 |- ( ( { A } C_ A /\ A e. { A } ) -> A e. A )
10	7, 8, 9	syl2an 477	. . . 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 423	. 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 369   = wceq 1370   e. wcel 1758   _Vcvv 3078   u. cun 3437   C_ wss 3439   {csn 3988   suc csuc 4832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642  ax-reg 7922

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-sn 3989  df-pr 3991  df-suc 4836

This theorem is referenced by: (None)