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Theorem elgch 9000
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 8999 . . . 4  |- GCH  =  ( Fin  u.  { y  |  A. x  -.  ( y  ~<  x  /\  x  ~<  ~P y
) } )
21eleq2i 2519 . . 3  |-  ( A  e. GCH 
<->  A  e.  ( Fin 
u.  { y  | 
A. x  -.  (
y  ~<  x  /\  x  ~<  ~P y ) } ) )
3 elun 3628 . . 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 249 . 2  |-  ( A  e. GCH 
<->  ( A  e.  Fin  \/  A  e.  { y  |  A. x  -.  ( y  ~<  x  /\  x  ~<  ~P y
) } ) )
5 breq1 4437 . . . . . . 7  |-  ( y  =  A  ->  (
y  ~<  x  <->  A  ~<  x ) )
6 pweq 3997 . . . . . . . 8  |-  ( y  =  A  ->  ~P y  =  ~P A
)
76breq2d 4446 . . . . . . 7  |-  ( y  =  A  ->  (
x  ~<  ~P y  <->  x  ~<  ~P A ) )
85, 7anbi12d 710 . . . . . 6  |-  ( y  =  A  ->  (
( y  ~<  x  /\  x  ~<  ~P y
)  <->  ( A  ~<  x  /\  x  ~<  ~P A
) ) )
98notbid 294 . . . . 5  |-  ( y  =  A  ->  ( -.  ( y  ~<  x  /\  x  ~<  ~P y
)  <->  -.  ( A  ~<  x  /\  x  ~<  ~P A ) ) )
109albidv 1698 . . . 4  |-  ( y  =  A  ->  ( A. x  -.  (
y  ~<  x  /\  x  ~<  ~P y )  <->  A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A
) ) )
1110elabg 3231 . . 3  |-  ( A  e.  V  ->  ( A  e.  { y  |  A. x  -.  (
y  ~<  x  /\  x  ~<  ~P y ) }  <->  A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A ) ) )
1211orbi2d 701 . 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 257 1  |-  ( A  e.  V  ->  ( A  e. GCH  <->  ( A  e. 
Fin  \/  A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A
) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369   A.wal 1379    = wceq 1381    e. wcel 1802   {cab 2426    u. cun 3457   ~Pcpw 3994   class class class wbr 4434    ~< csdm 7514   Fincfn 7515  GCHcgch 8998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-rab 2800  df-v 3095  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-op 4018  df-br 4435  df-gch 8999
This theorem is referenced by:  gchi  9002  engch  9006  hargch  9051  alephgch  9052
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