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Theorem elgch 8497
Description: Elementhood in the collection of GCH-sets. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
elgch  |-  ( A  e.  V  ->  ( A  e. GCH  <->  ( A  e. 
Fin  \/  A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A
) ) ) )
Distinct variable group:    x, A
Allowed substitution hint:    V( x)

Proof of Theorem elgch
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-gch 8496 . . . 4  |- GCH  =  ( Fin  u.  { y  |  A. x  -.  ( y  ~<  x  /\  x  ~<  ~P y
) } )
21eleq2i 2500 . . 3  |-  ( A  e. GCH 
<->  A  e.  ( Fin 
u.  { y  | 
A. x  -.  (
y  ~<  x  /\  x  ~<  ~P y ) } ) )
3 elun 3488 . . 3  |-  ( A  e.  ( Fin  u.  { y  |  A. x  -.  ( y  ~<  x  /\  x  ~<  ~P y
) } )  <->  ( A  e.  Fin  \/  A  e. 
{ y  |  A. x  -.  ( y  ~<  x  /\  x  ~<  ~P y
) } ) )
42, 3bitri 241 . 2  |-  ( A  e. GCH 
<->  ( A  e.  Fin  \/  A  e.  { y  |  A. x  -.  ( y  ~<  x  /\  x  ~<  ~P y
) } ) )
5 breq1 4215 . . . . . . 7  |-  ( y  =  A  ->  (
y  ~<  x  <->  A  ~<  x ) )
6 pweq 3802 . . . . . . . 8  |-  ( y  =  A  ->  ~P y  =  ~P A
)
76breq2d 4224 . . . . . . 7  |-  ( y  =  A  ->  (
x  ~<  ~P y  <->  x  ~<  ~P A ) )
85, 7anbi12d 692 . . . . . 6  |-  ( y  =  A  ->  (
( y  ~<  x  /\  x  ~<  ~P y
)  <->  ( A  ~<  x  /\  x  ~<  ~P A
) ) )
98notbid 286 . . . . 5  |-  ( y  =  A  ->  ( -.  ( y  ~<  x  /\  x  ~<  ~P y
)  <->  -.  ( A  ~<  x  /\  x  ~<  ~P A ) ) )
109albidv 1635 . . . 4  |-  ( y  =  A  ->  ( A. x  -.  (
y  ~<  x  /\  x  ~<  ~P y )  <->  A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A
) ) )
1110elabg 3083 . . 3  |-  ( A  e.  V  ->  ( A  e.  { y  |  A. x  -.  (
y  ~<  x  /\  x  ~<  ~P y ) }  <->  A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A ) ) )
1211orbi2d 683 . 2  |-  ( A  e.  V  ->  (
( A  e.  Fin  \/  A  e.  { y  |  A. x  -.  ( y  ~<  x  /\  x  ~<  ~P y
) } )  <->  ( A  e.  Fin  \/  A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A
) ) ) )
134, 12syl5bb 249 1  |-  ( A  e.  V  ->  ( A  e. GCH  <->  ( A  e. 
Fin  \/  A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A
) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359   A.wal 1549    = wceq 1652    e. wcel 1725   {cab 2422    u. cun 3318   ~Pcpw 3799   class class class wbr 4212    ~< csdm 7108   Fincfn 7109  GCHcgch 8495
This theorem is referenced by:  gchi  8499  engch  8503  hargch  8552  alephgch  8553
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-gch 8496
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