MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pwdom Structured version   Unicode version

Theorem pwdom 7463
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 3863 . . 3  |-  ( A  =  (/)  ->  ~P A  =  ~P (/) )
21breq1d 4302 . 2  |-  ( A  =  (/)  ->  ( ~P A  ~<_  ~P B  <->  ~P (/)  ~<_  ~P B
) )
3 reldom 7316 . . . . . . 7  |-  Rel  ~<_
43brrelexi 4879 . . . . . 6  |-  ( A  ~<_  B  ->  A  e.  _V )
5 0sdomg 7440 . . . . . 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 7462 . . . 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 2975 . . . . 5  |-  f  e. 
_V
12 fopwdom 7419 . . . . 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 1688 . . 3  |-  ( E. f  f : B -onto-> A  ->  ~P A  ~<_  ~P B )
1510, 14syl 16 . 2  |-  ( ( A  ~<_  B  /\  A  =/=  (/) )  ->  ~P A  ~<_  ~P B )
163brrelex2i 4880 . . . 4  |-  ( A  ~<_  B  ->  B  e.  _V )
17 pwexg 4476 . . . 4  |-  ( B  e.  _V  ->  ~P B  e.  _V )
1816, 17syl 16 . . 3  |-  ( A  ~<_  B  ->  ~P B  e.  _V )
19 0ss 3666 . . . 4  |-  (/)  C_  B
20 sspwb 4541 . . . 4  |-  ( (/)  C_  B  <->  ~P (/)  C_  ~P B
)
2119, 20mpbi 208 . . 3  |-  ~P (/)  C_  ~P B
22 ssdomg 7355 . . 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 2686 1  |-  ( A  ~<_  B  ->  ~P A  ~<_  ~P B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756    =/= wne 2606   _Vcvv 2972    C_ wss 3328   (/)c0 3637   ~Pcpw 3860   class class class wbr 4292   -onto->wfo 5416    ~<_ cdom 7308    ~< csdm 7309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313
This theorem is referenced by:  cdalepw  8365  gchpwdom  8837  gchaclem  8845  2ndcredom  19054
  Copyright terms: Public domain W3C validator