Theorem sucprcreg 8114

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 5498	. 2 |- ( -. A e. _V -> suc A = A )
2		elirr 8113	. . . 4 |- -. A e. A
3		df-suc 5429	. . . . . . . 8 |- suc A = ( A u. { A } )
4	3	eqeq1i 2456	. . . . . . 7 |- ( suc A = A <-> ( A u. { A } ) = A )
5		ssequn2 3607	. . . . . . 7 |- ( { A } C_ A <-> ( A u. { A } ) = A )
6	4, 5	bitr4i 256	. . . . . 6 |- ( suc A = A <-> { A } C_ A )
7	6	biimpi 198	. . . . 5 |- ( suc A = A -> { A } C_ A )
8		snidg 3994	. . . . 5 |- ( A e. _V -> A e. { A } )
9		ssel2 3427	. . . . 5 |- ( ( { A } C_ A /\ A e. { A } ) -> A e. A )
10	7, 8, 9	syl2an 480	. . . 4 |- ( ( suc A = A /\ A e. _V ) -> A e. A )
11	2, 10	mto 180	. . 3 |- -. ( suc A = A /\ A e. _V )
12	11	imnani 425	. 2 |- ( suc A = A -> -. A e. _V )
13	1, 12	impbii 191	1 |- ( -. A e. _V <-> suc A = A )

Syntax hints:   -. wn 3   <-> wb 188   /\ wa 371   = wceq 1444   e. wcel 1887   _Vcvv 3045   u. cun 3402   C_ wss 3404   {csn 3968   suc csuc 5425

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639  ax-reg 8107

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-sn 3969  df-pr 3971  df-suc 5429

This theorem is referenced by: (None)