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Theorem elgch 8244
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 8243 . . . 4  |- GCH  =  ( Fin  u.  { y  |  A. x  -.  ( y  ~<  x  /\  x  ~<  ~P y
) } )
21eleq2i 2347 . . 3  |-  ( A  e. GCH 
<->  A  e.  ( Fin 
u.  { y  | 
A. x  -.  (
y  ~<  x  /\  x  ~<  ~P y ) } ) )
3 elun 3316 . . 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 240 . 2  |-  ( A  e. GCH 
<->  ( A  e.  Fin  \/  A  e.  { y  |  A. x  -.  ( y  ~<  x  /\  x  ~<  ~P y
) } ) )
5 breq1 4026 . . . . . . 7  |-  ( y  =  A  ->  (
y  ~<  x  <->  A  ~<  x ) )
6 pweq 3628 . . . . . . . 8  |-  ( y  =  A  ->  ~P y  =  ~P A
)
76breq2d 4035 . . . . . . 7  |-  ( y  =  A  ->  (
x  ~<  ~P y  <->  x  ~<  ~P A ) )
85, 7anbi12d 691 . . . . . 6  |-  ( y  =  A  ->  (
( y  ~<  x  /\  x  ~<  ~P y
)  <->  ( A  ~<  x  /\  x  ~<  ~P A
) ) )
98notbid 285 . . . . 5  |-  ( y  =  A  ->  ( -.  ( y  ~<  x  /\  x  ~<  ~P y
)  <->  -.  ( A  ~<  x  /\  x  ~<  ~P A ) ) )
109albidv 1611 . . . 4  |-  ( y  =  A  ->  ( A. x  -.  (
y  ~<  x  /\  x  ~<  ~P y )  <->  A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A
) ) )
1110elabg 2915 . . 3  |-  ( A  e.  V  ->  ( A  e.  { y  |  A. x  -.  (
y  ~<  x  /\  x  ~<  ~P y ) }  <->  A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A ) ) )
1211orbi2d 682 . 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 248 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 176    \/ wo 357    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684   {cab 2269    u. cun 3150   ~Pcpw 3625   class class class wbr 4023    ~< csdm 6862   Fincfn 6863  GCHcgch 8242
This theorem is referenced by:  gchi  8246  engch  8250  hargch  8299  alephgch  8300
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-gch 8243
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