MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sucprcreg Structured version   Unicode version

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
StepHypRef 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 } )
43eqeq1i 2409 . . . . . . 7  |-  ( suc 
A  =  A  <->  ( A  u.  { A } )  =  A )
5 ssequn2 3615 . . . . . . 7  |-  ( { A }  C_  A  <->  ( A  u.  { A } )  =  A )
64, 5bitr4i 252 . . . . . 6  |-  ( suc 
A  =  A  <->  { A }  C_  A )
76biimpi 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 )
107, 8, 9syl2an 475 . . . 4  |-  ( ( suc  A  =  A  /\  A  e.  _V )  ->  A  e.  A
)
112, 10mto 176 . . 3  |-  -.  ( suc  A  =  A  /\  A  e.  _V )
1211imnani 421 . 2  |-  ( suc 
A  =  A  ->  -.  A  e.  _V )
131, 12impbii 188 1  |-  ( -.  A  e.  _V  <->  suc  A  =  A )
Colors of variables: wff setvar class
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)
  Copyright terms: Public domain W3C validator