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Theorem pwdom 7666
Description: Injection of sets implies injection on power sets. (Contributed by Mario Carneiro, 9-Apr-2015.)
Assertion
Ref Expression
pwdom  |-  ( A  ~<_  B  ->  ~P A  ~<_  ~P B )

Proof of Theorem pwdom
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 pweq 4013 . . 3  |-  ( A  =  (/)  ->  ~P A  =  ~P (/) )
21breq1d 4457 . 2  |-  ( A  =  (/)  ->  ( ~P A  ~<_  ~P B  <->  ~P (/)  ~<_  ~P B
) )
3 reldom 7519 . . . . . . 7  |-  Rel  ~<_
43brrelexi 5039 . . . . . 6  |-  ( A  ~<_  B  ->  A  e.  _V )
5 0sdomg 7643 . . . . . 6  |-  ( A  e.  _V  ->  ( (/) 
~<  A  <->  A  =/=  (/) ) )
64, 5syl 16 . . . . 5  |-  ( A  ~<_  B  ->  ( (/)  ~<  A  <->  A  =/=  (/) ) )
76biimpar 485 . . . 4  |-  ( ( A  ~<_  B  /\  A  =/=  (/) )  ->  (/)  ~<  A )
8 simpl 457 . . . 4  |-  ( ( A  ~<_  B  /\  A  =/=  (/) )  ->  A  ~<_  B )
9 fodomr 7665 . . . 4  |-  ( (
(/)  ~<  A  /\  A  ~<_  B )  ->  E. f 
f : B -onto-> A
)
107, 8, 9syl2anc 661 . . 3  |-  ( ( A  ~<_  B  /\  A  =/=  (/) )  ->  E. f 
f : B -onto-> A
)
11 vex 3116 . . . . 5  |-  f  e. 
_V
12 fopwdom 7622 . . . . 5  |-  ( ( f  e.  _V  /\  f : B -onto-> A )  ->  ~P A  ~<_  ~P B )
1311, 12mpan 670 . . . 4  |-  ( f : B -onto-> A  ->  ~P A  ~<_  ~P B
)
1413exlimiv 1698 . . 3  |-  ( E. f  f : B -onto-> A  ->  ~P A  ~<_  ~P B )
1510, 14syl 16 . 2  |-  ( ( A  ~<_  B  /\  A  =/=  (/) )  ->  ~P A  ~<_  ~P B )
163brrelex2i 5040 . . . 4  |-  ( A  ~<_  B  ->  B  e.  _V )
17 pwexg 4631 . . . 4  |-  ( B  e.  _V  ->  ~P B  e.  _V )
1816, 17syl 16 . . 3  |-  ( A  ~<_  B  ->  ~P B  e.  _V )
19 0ss 3814 . . . 4  |-  (/)  C_  B
20 sspwb 4696 . . . 4  |-  ( (/)  C_  B  <->  ~P (/)  C_  ~P B
)
2119, 20mpbi 208 . . 3  |-  ~P (/)  C_  ~P B
22 ssdomg 7558 . . 3  |-  ( ~P B  e.  _V  ->  ( ~P (/)  C_  ~P B  ->  ~P (/)  ~<_  ~P B
) )
2318, 21, 22mpisyl 18 . 2  |-  ( A  ~<_  B  ->  ~P (/)  ~<_  ~P B
)
242, 15, 23pm2.61ne 2782 1  |-  ( A  ~<_  B  ->  ~P A  ~<_  ~P B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767    =/= wne 2662   _Vcvv 3113    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   class class class wbr 4447   -onto->wfo 5584    ~<_ cdom 7511    ~< csdm 7512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516
This theorem is referenced by:  cdalepw  8572  gchpwdom  9044  gchaclem  9052  2ndcredom  19714
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