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Theorem engch 8437
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 7262 . . 3  |-  ( A 
~~  B  ->  ( A  e.  Fin  <->  B  e.  Fin ) )
2 sdomen1 7188 . . . . . 6  |-  ( A 
~~  B  ->  ( A  ~<  x  <->  B  ~<  x ) )
3 pwen 7217 . . . . . . 7  |-  ( A 
~~  B  ->  ~P A  ~~  ~P B )
4 sdomen2 7189 . . . . . . 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 692 . . . . 5  |-  ( A 
~~  B  ->  (
( A  ~<  x  /\  x  ~<  ~P A
)  <->  ( B  ~<  x  /\  x  ~<  ~P B
) ) )
76notbid 286 . . . 4  |-  ( A 
~~  B  ->  ( -.  ( A  ~<  x  /\  x  ~<  ~P A
)  <->  -.  ( B  ~<  x  /\  x  ~<  ~P B ) ) )
87albidv 1632 . . 3  |-  ( A 
~~  B  ->  ( A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A )  <->  A. x  -.  ( B  ~<  x  /\  x  ~<  ~P B
) ) )
91, 8orbi12d 691 . 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 7051 . . . 4  |-  Rel  ~~
1110brrelexi 4859 . . 3  |-  ( A 
~~  B  ->  A  e.  _V )
12 elgch 8431 . . 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 4860 . . 3  |-  ( A 
~~  B  ->  B  e.  _V )
15 elgch 8431 . . 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 277 1  |-  ( A 
~~  B  ->  ( A  e. GCH  <->  B  e. GCH )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359   A.wal 1546    e. wcel 1717   _Vcvv 2900   ~Pcpw 3743   class class class wbr 4154    ~~ cen 7043    ~< csdm 7045   Fincfn 7046  GCHcgch 8429
This theorem is referenced by:  gch2  8488
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-1o 6661  df-2o 6662  df-er 6842  df-map 6957  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-gch 8430
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