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Theorem pwcdaidm 8001
Description: If the natural numbers inject into  A, then  ~P A is idempotent under cardinal sum. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
pwcdaidm  |-  ( om  ~<_  A  ->  ( ~P A  +c  ~P A ) 
~~  ~P A )

Proof of Theorem pwcdaidm
StepHypRef Expression
1 reldom 7044 . . . 4  |-  Rel  ~<_
21brrelex2i 4852 . . 3  |-  ( om  ~<_  A  ->  A  e.  _V )
3 pwcda1 8000 . . 3  |-  ( A  e.  _V  ->  ( ~P A  +c  ~P A
)  ~~  ~P ( A  +c  1o ) )
42, 3syl 16 . 2  |-  ( om  ~<_  A  ->  ( ~P A  +c  ~P A ) 
~~  ~P ( A  +c  1o ) )
5 infcda1 7999 . . 3  |-  ( om  ~<_  A  ->  ( A  +c  1o )  ~~  A
)
6 pwen 7209 . . 3  |-  ( ( A  +c  1o ) 
~~  A  ->  ~P ( A  +c  1o )  ~~  ~P A )
75, 6syl 16 . 2  |-  ( om  ~<_  A  ->  ~P ( A  +c  1o )  ~~  ~P A )
8 entr 7088 . 2  |-  ( ( ( ~P A  +c  ~P A )  ~~  ~P ( A  +c  1o )  /\  ~P ( A  +c  1o )  ~~  ~P A )  ->  ( ~P A  +c  ~P A
)  ~~  ~P A
)
94, 7, 8syl2anc 643 1  |-  ( om  ~<_  A  ->  ( ~P A  +c  ~P A ) 
~~  ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1717   _Vcvv 2892   ~Pcpw 3735   class class class wbr 4146   omcom 4778  (class class class)co 6013   1oc1o 6646    ~~ cen 7035    ~<_ cdom 7036    +c ccda 7973
This theorem is referenced by:  gchaclem  8471
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 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-1o 6653  df-2o 6654  df-er 6834  df-map 6949  df-en 7039  df-dom 7040  df-cda 7974
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