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Theorem pwcda1 7836
Description: The sum of a powerset with itself is equipotent to the successor powerset. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
pwcda1  |-  ( A  e.  V  ->  ( ~P A  +c  ~P A
)  ~~  ~P ( A  +c  1o ) )

Proof of Theorem pwcda1
StepHypRef Expression
1 1on 6502 . . . 4  |-  1o  e.  On
2 pwcdaen 7827 . . . 4  |-  ( ( A  e.  V  /\  1o  e.  On )  ->  ~P ( A  +c  1o )  ~~  ( ~P A  X.  ~P 1o ) )
31, 2mpan2 652 . . 3  |-  ( A  e.  V  ->  ~P ( A  +c  1o )  ~~  ( ~P A  X.  ~P 1o ) )
4 pwpw0 3779 . . . . . 6  |-  ~P { (/)
}  =  { (/) ,  { (/) } }
5 df1o2 6507 . . . . . . 7  |-  1o  =  { (/) }
65pweqi 3642 . . . . . 6  |-  ~P 1o  =  ~P { (/) }
7 df2o2 6509 . . . . . 6  |-  2o  =  { (/) ,  { (/) } }
84, 6, 73eqtr4i 2326 . . . . 5  |-  ~P 1o  =  2o
98xpeq2i 4726 . . . 4  |-  ( ~P A  X.  ~P 1o )  =  ( ~P A  X.  2o )
10 pwexg 4210 . . . . 5  |-  ( A  e.  V  ->  ~P A  e.  _V )
11 xp2cda 7822 . . . . 5  |-  ( ~P A  e.  _V  ->  ( ~P A  X.  2o )  =  ( ~P A  +c  ~P A ) )
1210, 11syl 15 . . . 4  |-  ( A  e.  V  ->  ( ~P A  X.  2o )  =  ( ~P A  +c  ~P A ) )
139, 12syl5eq 2340 . . 3  |-  ( A  e.  V  ->  ( ~P A  X.  ~P 1o )  =  ( ~P A  +c  ~P A ) )
143, 13breqtrd 4063 . 2  |-  ( A  e.  V  ->  ~P ( A  +c  1o )  ~~  ( ~P A  +c  ~P A ) )
15 ensym 6926 . 2  |-  ( ~P ( A  +c  1o )  ~~  ( ~P A  +c  ~P A )  -> 
( ~P A  +c  ~P A )  ~~  ~P ( A  +c  1o ) )
1614, 15syl 15 1  |-  ( A  e.  V  ->  ( ~P A  +c  ~P A
)  ~~  ~P ( A  +c  1o ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468   ~Pcpw 3638   {csn 3653   {cpr 3654   class class class wbr 4039   Oncon0 4408    X. cxp 4703  (class class class)co 5874   1oc1o 6488   2oc2o 6489    ~~ cen 6876    +c ccda 7809
This theorem is referenced by:  pwcdaidm  7837  cdalepw  7838  pwsdompw  7846  gchcdaidm  8306  gchpwdom  8312
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-int 3879  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-suc 4414  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-cda 7810
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