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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
StepHypRef 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 } )
43eqeq1i 2456 . . . . . . 7  |-  ( suc 
A  =  A  <->  ( A  u.  { A } )  =  A )
5 ssequn2 3607 . . . . . . 7  |-  ( { A }  C_  A  <->  ( A  u.  { A } )  =  A )
64, 5bitr4i 256 . . . . . 6  |-  ( suc 
A  =  A  <->  { A }  C_  A )
76biimpi 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 )
107, 8, 9syl2an 480 . . . 4  |-  ( ( suc  A  =  A  /\  A  e.  _V )  ->  A  e.  A
)
112, 10mto 180 . . 3  |-  -.  ( suc  A  =  A  /\  A  e.  _V )
1211imnani 425 . 2  |-  ( suc 
A  =  A  ->  -.  A  e.  _V )
131, 12impbii 191 1  |-  ( -.  A  e.  _V  <->  suc  A  =  A )
Colors of variables: wff setvar class
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)
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