Theorem sucprcreg 8506

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 5800	. 2 |- ( -. A e. _V -> suc A = A )
2		elirr 8505	. . . 4 |- -. A e. A
3		df-suc 5729	. . . . . . . 8 |- suc A = ( A u. { A } )
4	3	eqeq1i 2627	. . . . . . 7 |- ( suc A = A <-> ( A u. { A } ) = A )
5		ssequn2 3786	. . . . . . 7 |- ( { A } C_ A <-> ( A u. { A } ) = A )
6	4, 5	bitr4i 267	. . . . . 6 |- ( suc A = A <-> { A } C_ A )
7	6	biimpi 206	. . . . 5 |- ( suc A = A -> { A } C_ A )
8		snidg 4206	. . . . 5 |- ( A e. _V -> A e. { A } )
9		ssel2 3598	. . . . 5 |- ( ( { A } C_ A /\ A e. { A } ) -> A e. A )
10	7, 8, 9	syl2an 494	. . . 4 |- ( ( suc A = A /\ A e. _V ) -> A e. A )
11	2, 10	mto 188	. . 3 |- -. ( suc A = A /\ A e. _V )
12	11	imnani 439	. 2 |- ( suc A = A -> -. A e. _V )
13	1, 12	impbii 199	1 |- ( -. A e. _V <-> suc A = A )

Syntax hints:   -. wn 3   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   C_ wss 3574   {csn 4177   suc csuc 5725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-reg 8497

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180  df-suc 5729

This theorem is referenced by: (None)