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Theorem pwdom 7013
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 3628 . . 3  |-  ( A  =  (/)  ->  ~P A  =  ~P (/) )
21breq1d 4033 . 2  |-  ( A  =  (/)  ->  ( ~P A  ~<_  ~P B  <->  ~P (/)  ~<_  ~P B
) )
3 reldom 6869 . . . . . . 7  |-  Rel  ~<_
43brrelexi 4729 . . . . . 6  |-  ( A  ~<_  B  ->  A  e.  _V )
5 0sdomg 6990 . . . . . 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 7012 . . . 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 2791 . . . . 5  |-  f  e. 
_V
12 fopwdom 6970 . . . . 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 1666 . . 3  |-  ( E. f  f : B -onto-> A  ->  ~P A  ~<_  ~P B )
1510, 14syl 15 . 2  |-  ( ( A  ~<_  B  /\  A  =/=  (/) )  ->  ~P A  ~<_  ~P B )
163brrelex2i 4730 . . . 4  |-  ( A  ~<_  B  ->  B  e.  _V )
17 pwexg 4194 . . . 4  |-  ( B  e.  _V  ->  ~P B  e.  _V )
1816, 17syl 15 . . 3  |-  ( A  ~<_  B  ->  ~P B  e.  _V )
19 0ss 3483 . . . 4  |-  (/)  C_  B
20 sspwb 4223 . . . 4  |-  ( (/)  C_  B  <->  ~P (/)  C_  ~P B
)
2119, 20mpbi 199 . . 3  |-  ~P (/)  C_  ~P B
22 ssdomg 6907 . . 3  |-  ( ~P B  e.  _V  ->  ( ~P (/)  C_  ~P B  ->  ~P (/)  ~<_  ~P B
) )
2318, 21, 22ee10 1366 . 2  |-  ( A  ~<_  B  ->  ~P (/)  ~<_  ~P B
)
242, 15, 23pm2.61ne 2521 1  |-  ( A  ~<_  B  ->  ~P A  ~<_  ~P B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   class class class wbr 4023   -onto->wfo 5253    ~<_ cdom 6861    ~< csdm 6862
This theorem is referenced by:  cdalepw  7822  gchaclem  8292  gchpwdom  8296  2ndcredom  17176
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866
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