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Theorem engch 8816
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 7550 . . 3  |-  ( A 
~~  B  ->  ( A  e.  Fin  <->  B  e.  Fin ) )
2 sdomen1 7476 . . . . . 6  |-  ( A 
~~  B  ->  ( A  ~<  x  <->  B  ~<  x ) )
3 pwen 7505 . . . . . . 7  |-  ( A 
~~  B  ->  ~P A  ~~  ~P B )
4 sdomen2 7477 . . . . . . 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 1679 . . 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 7336 . . . 4  |-  Rel  ~~
1110brrelexi 4900 . . 3  |-  ( A 
~~  B  ->  A  e.  _V )
12 elgch 8810 . . 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 4901 . . 3  |-  ( A 
~~  B  ->  B  e.  _V )
15 elgch 8810 . . 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 1367    e. wcel 1756   _Vcvv 2993   ~Pcpw 3881   class class class wbr 4313    ~~ cen 7328    ~< csdm 7330   Fincfn 7331  GCHcgch 8808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-1o 6941  df-2o 6942  df-er 7122  df-map 7237  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-gch 8809
This theorem is referenced by:  gch2  8863
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