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Theorem engch 9018
Description: The property of being a GCH-set is a cardinal invariant. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
engch  |-  ( A 
~~  B  ->  ( A  e. GCH  <->  B  e. GCH )
)

Proof of Theorem engch
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 enfi 7748 . . 3  |-  ( A 
~~  B  ->  ( A  e.  Fin  <->  B  e.  Fin ) )
2 sdomen1 7673 . . . . . 6  |-  ( A 
~~  B  ->  ( A  ~<  x  <->  B  ~<  x ) )
3 pwen 7702 . . . . . . 7  |-  ( A 
~~  B  ->  ~P A  ~~  ~P B )
4 sdomen2 7674 . . . . . . 7  |-  ( ~P A  ~~  ~P B  ->  ( x  ~<  ~P A  <->  x 
~<  ~P B ) )
53, 4syl 16 . . . . . 6  |-  ( A 
~~  B  ->  (
x  ~<  ~P A  <->  x  ~<  ~P B ) )
62, 5anbi12d 710 . . . . 5  |-  ( A 
~~  B  ->  (
( A  ~<  x  /\  x  ~<  ~P A
)  <->  ( B  ~<  x  /\  x  ~<  ~P B
) ) )
76notbid 294 . . . 4  |-  ( A 
~~  B  ->  ( -.  ( A  ~<  x  /\  x  ~<  ~P A
)  <->  -.  ( B  ~<  x  /\  x  ~<  ~P B ) ) )
87albidv 1689 . . 3  |-  ( A 
~~  B  ->  ( A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A )  <->  A. x  -.  ( B  ~<  x  /\  x  ~<  ~P B
) ) )
91, 8orbi12d 709 . 2  |-  ( A 
~~  B  ->  (
( A  e.  Fin  \/ 
A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A ) )  <-> 
( B  e.  Fin  \/ 
A. x  -.  ( B  ~<  x  /\  x  ~<  ~P B ) ) ) )
10 relen 7533 . . . 4  |-  Rel  ~~
1110brrelexi 5046 . . 3  |-  ( A 
~~  B  ->  A  e.  _V )
12 elgch 9012 . . 3  |-  ( A  e.  _V  ->  ( A  e. GCH  <->  ( A  e. 
Fin  \/  A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A
) ) ) )
1311, 12syl 16 . 2  |-  ( A 
~~  B  ->  ( A  e. GCH  <->  ( A  e. 
Fin  \/  A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A
) ) ) )
1410brrelex2i 5047 . . 3  |-  ( A 
~~  B  ->  B  e.  _V )
15 elgch 9012 . . 3  |-  ( B  e.  _V  ->  ( B  e. GCH  <->  ( B  e. 
Fin  \/  A. x  -.  ( B  ~<  x  /\  x  ~<  ~P B
) ) ) )
1614, 15syl 16 . 2  |-  ( A 
~~  B  ->  ( B  e. GCH  <->  ( B  e. 
Fin  \/  A. x  -.  ( B  ~<  x  /\  x  ~<  ~P B
) ) ) )
179, 13, 163bitr4d 285 1  |-  ( A 
~~  B  ->  ( A  e. GCH  <->  B  e. GCH )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369   A.wal 1377    e. wcel 1767   _Vcvv 3118   ~Pcpw 4016   class class class wbr 4453    ~~ cen 7525    ~< csdm 7527   Fincfn 7528  GCHcgch 9010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-1o 7142  df-2o 7143  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-gch 9011
This theorem is referenced by:  gch2  9065
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