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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
StepHypRef 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 } )
43eqeq1i 2461 . . . . . . 7  |-  ( suc 
A  =  A  <->  ( A  u.  { A } )  =  A )
5 ssequn2 3640 . . . . . . 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 4014 . . . . 5  |-  ( A  e.  _V  ->  A  e.  { A } )
9 ssel2 3462 . . . . 5  |-  ( ( { A }  C_  A  /\  A  e.  { A } )  ->  A  e.  A )
107, 8, 9syl2an 477 . . . 4  |-  ( ( suc  A  =  A  /\  A  e.  _V )  ->  A  e.  A
)
112, 10mto 176 . . 3  |-  -.  ( suc  A  =  A  /\  A  e.  _V )
1211imnani 423 . 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 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)
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