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Theorem sucprcreg 8112
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 5509 . 2  |-  ( -.  A  e.  _V  ->  suc 
A  =  A )
2 elirr 8111 . . . 4  |-  -.  A  e.  A
3 df-suc 5440 . . . . . . . 8  |-  suc  A  =  ( A  u.  { A } )
43eqeq1i 2427 . . . . . . 7  |-  ( suc 
A  =  A  <->  ( A  u.  { A } )  =  A )
5 ssequn2 3636 . . . . . . 7  |-  ( { A }  C_  A  <->  ( A  u.  { A } )  =  A )
64, 5bitr4i 255 . . . . . 6  |-  ( suc 
A  =  A  <->  { A }  C_  A )
76biimpi 197 . . . . 5  |-  ( suc 
A  =  A  ->  { A }  C_  A
)
8 snidg 4019 . . . . 5  |-  ( A  e.  _V  ->  A  e.  { A } )
9 ssel2 3456 . . . . 5  |-  ( ( { A }  C_  A  /\  A  e.  { A } )  ->  A  e.  A )
107, 8, 9syl2an 479 . . . 4  |-  ( ( suc  A  =  A  /\  A  e.  _V )  ->  A  e.  A
)
112, 10mto 179 . . 3  |-  -.  ( suc  A  =  A  /\  A  e.  _V )
1211imnani 424 . 2  |-  ( suc 
A  =  A  ->  -.  A  e.  _V )
131, 12impbii 190 1  |-  ( -.  A  e.  _V  <->  suc  A  =  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1867   _Vcvv 3078    u. cun 3431    C_ wss 3433   {csn 3993   suc csuc 5436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4540  ax-nul 4548  ax-pr 4653  ax-reg 8105
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-sn 3994  df-pr 3996  df-suc 5440
This theorem is referenced by: (None)
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