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Theorem pwcda1 8361
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 6925 . . . 4  |-  1o  e.  On
2 pwcdaen 8352 . . . 4  |-  ( ( A  e.  V  /\  1o  e.  On )  ->  ~P ( A  +c  1o )  ~~  ( ~P A  X.  ~P 1o ) )
31, 2mpan2 671 . . 3  |-  ( A  e.  V  ->  ~P ( A  +c  1o )  ~~  ( ~P A  X.  ~P 1o ) )
4 pwpw0 4019 . . . . . 6  |-  ~P { (/)
}  =  { (/) ,  { (/) } }
5 df1o2 6930 . . . . . . 7  |-  1o  =  { (/) }
65pweqi 3862 . . . . . 6  |-  ~P 1o  =  ~P { (/) }
7 df2o2 6932 . . . . . 6  |-  2o  =  { (/) ,  { (/) } }
84, 6, 73eqtr4i 2471 . . . . 5  |-  ~P 1o  =  2o
98xpeq2i 4859 . . . 4  |-  ( ~P A  X.  ~P 1o )  =  ( ~P A  X.  2o )
10 pwexg 4474 . . . . 5  |-  ( A  e.  V  ->  ~P A  e.  _V )
11 xp2cda 8347 . . . . 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 2485 . . 3  |-  ( A  e.  V  ->  ( ~P A  X.  ~P 1o )  =  ( ~P A  +c  ~P A ) )
143, 13breqtrd 4314 . 2  |-  ( A  e.  V  ->  ~P ( A  +c  1o )  ~~  ( ~P A  +c  ~P A ) )
1514ensymd 7358 1  |-  ( A  e.  V  ->  ( ~P A  +c  ~P A
)  ~~  ~P ( A  +c  1o ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   _Vcvv 2970   (/)c0 3635   ~Pcpw 3858   {csn 3875   {cpr 3877   class class class wbr 4290   Oncon0 4717    X. cxp 4836  (class class class)co 6089   1oc1o 6911   2oc2o 6912    ~~ cen 7305    +c ccda 8334
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 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
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 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-1st 6575  df-2nd 6576  df-1o 6918  df-2o 6919  df-er 7099  df-map 7214  df-en 7309  df-dom 7310  df-cda 8335
This theorem is referenced by:  pwcdaidm  8362  cdalepw  8363  pwsdompw  8371  gchcdaidm  8833  gchpwdom  8835
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