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Theorem pwcda1 8008
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 6668 . . . 4  |-  1o  e.  On
2 pwcdaen 7999 . . . 4  |-  ( ( A  e.  V  /\  1o  e.  On )  ->  ~P ( A  +c  1o )  ~~  ( ~P A  X.  ~P 1o ) )
31, 2mpan2 653 . . 3  |-  ( A  e.  V  ->  ~P ( A  +c  1o )  ~~  ( ~P A  X.  ~P 1o ) )
4 pwpw0 3890 . . . . . 6  |-  ~P { (/)
}  =  { (/) ,  { (/) } }
5 df1o2 6673 . . . . . . 7  |-  1o  =  { (/) }
65pweqi 3747 . . . . . 6  |-  ~P 1o  =  ~P { (/) }
7 df2o2 6675 . . . . . 6  |-  2o  =  { (/) ,  { (/) } }
84, 6, 73eqtr4i 2418 . . . . 5  |-  ~P 1o  =  2o
98xpeq2i 4840 . . . 4  |-  ( ~P A  X.  ~P 1o )  =  ( ~P A  X.  2o )
10 pwexg 4325 . . . . 5  |-  ( A  e.  V  ->  ~P A  e.  _V )
11 xp2cda 7994 . . . . 5  |-  ( ~P A  e.  _V  ->  ( ~P A  X.  2o )  =  ( ~P A  +c  ~P A ) )
1210, 11syl 16 . . . 4  |-  ( A  e.  V  ->  ( ~P A  X.  2o )  =  ( ~P A  +c  ~P A ) )
139, 12syl5eq 2432 . . 3  |-  ( A  e.  V  ->  ( ~P A  X.  ~P 1o )  =  ( ~P A  +c  ~P A ) )
143, 13breqtrd 4178 . 2  |-  ( A  e.  V  ->  ~P ( A  +c  1o )  ~~  ( ~P A  +c  ~P A ) )
1514ensymd 7095 1  |-  ( A  e.  V  ->  ( ~P A  +c  ~P A
)  ~~  ~P ( A  +c  1o ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   _Vcvv 2900   (/)c0 3572   ~Pcpw 3743   {csn 3758   {cpr 3759   class class class wbr 4154   Oncon0 4523    X. cxp 4817  (class class class)co 6021   1oc1o 6654   2oc2o 6655    ~~ cen 7043    +c ccda 7981
This theorem is referenced by:  pwcdaidm  8009  cdalepw  8010  pwsdompw  8018  gchcdaidm  8477  gchpwdom  8483
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-suc 4529  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-1o 6661  df-2o 6662  df-er 6842  df-map 6957  df-en 7047  df-dom 7048  df-cda 7982
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