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Theorem pwcda1 8622
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 7197 . . . 4  |-  1o  e.  On
2 pwcdaen 8613 . . . 4  |-  ( ( A  e.  V  /\  1o  e.  On )  ->  ~P ( A  +c  1o )  ~~  ( ~P A  X.  ~P 1o ) )
31, 2mpan2 675 . . 3  |-  ( A  e.  V  ->  ~P ( A  +c  1o )  ~~  ( ~P A  X.  ~P 1o ) )
4 pwpw0 4151 . . . . . 6  |-  ~P { (/)
}  =  { (/) ,  { (/) } }
5 df1o2 7202 . . . . . . 7  |-  1o  =  { (/) }
65pweqi 3989 . . . . . 6  |-  ~P 1o  =  ~P { (/) }
7 df2o2 7204 . . . . . 6  |-  2o  =  { (/) ,  { (/) } }
84, 6, 73eqtr4i 2468 . . . . 5  |-  ~P 1o  =  2o
98xpeq2i 4875 . . . 4  |-  ( ~P A  X.  ~P 1o )  =  ( ~P A  X.  2o )
10 pwexg 4609 . . . . 5  |-  ( A  e.  V  ->  ~P A  e.  _V )
11 xp2cda 8608 . . . . 5  |-  ( ~P A  e.  _V  ->  ( ~P A  X.  2o )  =  ( ~P A  +c  ~P A ) )
1210, 11syl 17 . . . 4  |-  ( A  e.  V  ->  ( ~P A  X.  2o )  =  ( ~P A  +c  ~P A ) )
139, 12syl5eq 2482 . . 3  |-  ( A  e.  V  ->  ( ~P A  X.  ~P 1o )  =  ( ~P A  +c  ~P A ) )
143, 13breqtrd 4450 . 2  |-  ( A  e.  V  ->  ~P ( A  +c  1o )  ~~  ( ~P A  +c  ~P A ) )
1514ensymd 7627 1  |-  ( A  e.  V  ->  ( ~P A  +c  ~P A
)  ~~  ~P ( A  +c  1o ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1870   _Vcvv 3087   (/)c0 3767   ~Pcpw 3985   {csn 4002   {cpr 4004   class class class wbr 4426    X. cxp 4852   Oncon0 5442  (class class class)co 6305   1oc1o 7183   2oc2o 7184    ~~ cen 7574    +c ccda 8595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-ord 5445  df-on 5446  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-1o 7190  df-2o 7191  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-cda 8596
This theorem is referenced by:  pwcdaidm  8623  cdalepw  8624  pwsdompw  8632  gchcdaidm  9092  gchpwdom  9094
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