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Theorem engch 8250
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 7079 . . 3  |-  ( A 
~~  B  ->  ( A  e.  Fin  <->  B  e.  Fin ) )
2 sdomen1 7005 . . . . . 6  |-  ( A 
~~  B  ->  ( A  ~<  x  <->  B  ~<  x ) )
3 pwen 7034 . . . . . . 7  |-  ( A 
~~  B  ->  ~P A  ~~  ~P B )
4 sdomen2 7006 . . . . . . 7  |-  ( ~P A  ~~  ~P B  ->  ( x  ~<  ~P A  <->  x 
~<  ~P B ) )
53, 4syl 15 . . . . . 6  |-  ( A 
~~  B  ->  (
x  ~<  ~P A  <->  x  ~<  ~P B ) )
62, 5anbi12d 691 . . . . 5  |-  ( A 
~~  B  ->  (
( A  ~<  x  /\  x  ~<  ~P A
)  <->  ( B  ~<  x  /\  x  ~<  ~P B
) ) )
76notbid 285 . . . 4  |-  ( A 
~~  B  ->  ( -.  ( A  ~<  x  /\  x  ~<  ~P A
)  <->  -.  ( B  ~<  x  /\  x  ~<  ~P B ) ) )
87albidv 1611 . . 3  |-  ( A 
~~  B  ->  ( A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A )  <->  A. x  -.  ( B  ~<  x  /\  x  ~<  ~P B
) ) )
91, 8orbi12d 690 . 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 6868 . . . 4  |-  Rel  ~~
1110brrelexi 4729 . . 3  |-  ( A 
~~  B  ->  A  e.  _V )
12 elgch 8244 . . 3  |-  ( A  e.  _V  ->  ( A  e. GCH  <->  ( A  e. 
Fin  \/  A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A
) ) ) )
1311, 12syl 15 . 2  |-  ( A 
~~  B  ->  ( A  e. GCH  <->  ( A  e. 
Fin  \/  A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A
) ) ) )
1410brrelex2i 4730 . . 3  |-  ( A 
~~  B  ->  B  e.  _V )
15 elgch 8244 . . 3  |-  ( B  e.  _V  ->  ( B  e. GCH  <->  ( B  e. 
Fin  \/  A. x  -.  ( B  ~<  x  /\  x  ~<  ~P B
) ) ) )
1614, 15syl 15 . 2  |-  ( A 
~~  B  ->  ( B  e. GCH  <->  ( B  e. 
Fin  \/  A. x  -.  ( B  ~<  x  /\  x  ~<  ~P B
) ) ) )
179, 13, 163bitr4d 276 1  |-  ( A 
~~  B  ->  ( A  e. GCH  <->  B  e. GCH )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358   A.wal 1527    e. wcel 1684   _Vcvv 2788   ~Pcpw 3625   class class class wbr 4023    ~~ cen 6860    ~< csdm 6862   Fincfn 6863  GCHcgch 8242
This theorem is referenced by:  gch2  8301
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-1o 6479  df-2o 6480  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-gch 8243
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