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Theorem pwdom 7056
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 3662 . . 3  |-  ( A  =  (/)  ->  ~P A  =  ~P (/) )
21breq1d 4070 . 2  |-  ( A  =  (/)  ->  ( ~P A  ~<_  ~P B  <->  ~P (/)  ~<_  ~P B
) )
3 reldom 6912 . . . . . . 7  |-  Rel  ~<_
43brrelexi 4766 . . . . . 6  |-  ( A  ~<_  B  ->  A  e.  _V )
5 0sdomg 7033 . . . . . 6  |-  ( A  e.  _V  ->  ( (/) 
~<  A  <->  A  =/=  (/) ) )
64, 5syl 15 . . . . 5  |-  ( A  ~<_  B  ->  ( (/)  ~<  A  <->  A  =/=  (/) ) )
76biimpar 471 . . . 4  |-  ( ( A  ~<_  B  /\  A  =/=  (/) )  ->  (/)  ~<  A )
8 simpl 443 . . . 4  |-  ( ( A  ~<_  B  /\  A  =/=  (/) )  ->  A  ~<_  B )
9 fodomr 7055 . . . 4  |-  ( (
(/)  ~<  A  /\  A  ~<_  B )  ->  E. f 
f : B -onto-> A
)
107, 8, 9syl2anc 642 . . 3  |-  ( ( A  ~<_  B  /\  A  =/=  (/) )  ->  E. f 
f : B -onto-> A
)
11 vex 2825 . . . . 5  |-  f  e. 
_V
12 fopwdom 7013 . . . . 5  |-  ( ( f  e.  _V  /\  f : B -onto-> A )  ->  ~P A  ~<_  ~P B )
1311, 12mpan 651 . . . 4  |-  ( f : B -onto-> A  ->  ~P A  ~<_  ~P B
)
1413exlimiv 1625 . . 3  |-  ( E. f  f : B -onto-> A  ->  ~P A  ~<_  ~P B )
1510, 14syl 15 . 2  |-  ( ( A  ~<_  B  /\  A  =/=  (/) )  ->  ~P A  ~<_  ~P B )
163brrelex2i 4767 . . . 4  |-  ( A  ~<_  B  ->  B  e.  _V )
17 pwexg 4231 . . . 4  |-  ( B  e.  _V  ->  ~P B  e.  _V )
1816, 17syl 15 . . 3  |-  ( A  ~<_  B  ->  ~P B  e.  _V )
19 0ss 3517 . . . 4  |-  (/)  C_  B
20 sspwb 4260 . . . 4  |-  ( (/)  C_  B  <->  ~P (/)  C_  ~P B
)
2119, 20mpbi 199 . . 3  |-  ~P (/)  C_  ~P B
22 ssdomg 6950 . . 3  |-  ( ~P B  e.  _V  ->  ( ~P (/)  C_  ~P B  ->  ~P (/)  ~<_  ~P B
) )
2318, 21, 22ee10 1367 . 2  |-  ( A  ~<_  B  ->  ~P (/)  ~<_  ~P B
)
242, 15, 23pm2.61ne 2554 1  |-  ( A  ~<_  B  ->  ~P A  ~<_  ~P B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1532    = wceq 1633    e. wcel 1701    =/= wne 2479   _Vcvv 2822    C_ wss 3186   (/)c0 3489   ~Pcpw 3659   class class class wbr 4060   -onto->wfo 5290    ~<_ cdom 6904    ~< csdm 6905
This theorem is referenced by:  cdalepw  7867  gchaclem  8337  gchpwdom  8341  2ndcredom  17232
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909
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