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Theorem engch 8266
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 7095 . . 3  |-  ( A 
~~  B  ->  ( A  e.  Fin  <->  B  e.  Fin ) )
2 sdomen1 7021 . . . . . 6  |-  ( A 
~~  B  ->  ( A  ~<  x  <->  B  ~<  x ) )
3 pwen 7050 . . . . . . 7  |-  ( A 
~~  B  ->  ~P A  ~~  ~P B )
4 sdomen2 7022 . . . . . . 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 1615 . . 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 6884 . . . 4  |-  Rel  ~~
1110brrelexi 4745 . . 3  |-  ( A 
~~  B  ->  A  e.  _V )
12 elgch 8260 . . 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 4746 . . 3  |-  ( A 
~~  B  ->  B  e.  _V )
15 elgch 8260 . . 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 1530    e. wcel 1696   _Vcvv 2801   ~Pcpw 3638   class class class wbr 4039    ~~ cen 6876    ~< csdm 6878   Fincfn 6879  GCHcgch 8258
This theorem is referenced by:  gch2  8317
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-1o 6495  df-2o 6496  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-gch 8259
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