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Theorem pwcda1 7820
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 6486 . . . 4  |-  1o  e.  On
2 pwcdaen 7811 . . . 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 3763 . . . . . 6  |-  ~P { (/)
}  =  { (/) ,  { (/) } }
5 df1o2 6491 . . . . . . 7  |-  1o  =  { (/) }
65pweqi 3629 . . . . . 6  |-  ~P 1o  =  ~P { (/) }
7 df2o2 6493 . . . . . 6  |-  2o  =  { (/) ,  { (/) } }
84, 6, 73eqtr4i 2313 . . . . 5  |-  ~P 1o  =  2o
98xpeq2i 4710 . . . 4  |-  ( ~P A  X.  ~P 1o )  =  ( ~P A  X.  2o )
10 pwexg 4194 . . . . 5  |-  ( A  e.  V  ->  ~P A  e.  _V )
11 xp2cda 7806 . . . . 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 2327 . . 3  |-  ( A  e.  V  ->  ( ~P A  X.  ~P 1o )  =  ( ~P A  +c  ~P A ) )
143, 13breqtrd 4047 . 2  |-  ( A  e.  V  ->  ~P ( A  +c  1o )  ~~  ( ~P A  +c  ~P A ) )
15 ensym 6910 . 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 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   ~Pcpw 3625   {csn 3640   {cpr 3641   class class class wbr 4023   Oncon0 4392    X. cxp 4687  (class class class)co 5858   1oc1o 6472   2oc2o 6473    ~~ cen 6860    +c ccda 7793
This theorem is referenced by:  pwcdaidm  7821  cdalepw  7822  pwsdompw  7830  gchcdaidm  8290  gchpwdom  8296
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-int 3863  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-suc 4398  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-cda 7794
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