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Theorem pwdom 7627
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 3957 . . 3  |-  ( A  =  (/)  ->  ~P A  =  ~P (/) )
21breq1d 4404 . 2  |-  ( A  =  (/)  ->  ( ~P A  ~<_  ~P B  <->  ~P (/)  ~<_  ~P B
) )
3 reldom 7480 . . . . . . 7  |-  Rel  ~<_
43brrelexi 4983 . . . . . 6  |-  ( A  ~<_  B  ->  A  e.  _V )
5 0sdomg 7604 . . . . . 6  |-  ( A  e.  _V  ->  ( (/) 
~<  A  <->  A  =/=  (/) ) )
64, 5syl 17 . . . . 5  |-  ( A  ~<_  B  ->  ( (/)  ~<  A  <->  A  =/=  (/) ) )
76biimpar 483 . . . 4  |-  ( ( A  ~<_  B  /\  A  =/=  (/) )  ->  (/)  ~<  A )
8 simpl 455 . . . 4  |-  ( ( A  ~<_  B  /\  A  =/=  (/) )  ->  A  ~<_  B )
9 fodomr 7626 . . . 4  |-  ( (
(/)  ~<  A  /\  A  ~<_  B )  ->  E. f 
f : B -onto-> A
)
107, 8, 9syl2anc 659 . . 3  |-  ( ( A  ~<_  B  /\  A  =/=  (/) )  ->  E. f 
f : B -onto-> A
)
11 vex 3061 . . . . 5  |-  f  e. 
_V
12 fopwdom 7583 . . . . 5  |-  ( ( f  e.  _V  /\  f : B -onto-> A )  ->  ~P A  ~<_  ~P B )
1311, 12mpan 668 . . . 4  |-  ( f : B -onto-> A  ->  ~P A  ~<_  ~P B
)
1413exlimiv 1743 . . 3  |-  ( E. f  f : B -onto-> A  ->  ~P A  ~<_  ~P B )
1510, 14syl 17 . 2  |-  ( ( A  ~<_  B  /\  A  =/=  (/) )  ->  ~P A  ~<_  ~P B )
163brrelex2i 4984 . . . 4  |-  ( A  ~<_  B  ->  B  e.  _V )
17 pwexg 4577 . . . 4  |-  ( B  e.  _V  ->  ~P B  e.  _V )
1816, 17syl 17 . . 3  |-  ( A  ~<_  B  ->  ~P B  e.  _V )
19 0ss 3767 . . . 4  |-  (/)  C_  B
20 sspwb 4639 . . . 4  |-  ( (/)  C_  B  <->  ~P (/)  C_  ~P B
)
2119, 20mpbi 208 . . 3  |-  ~P (/)  C_  ~P B
22 ssdomg 7519 . . 3  |-  ( ~P B  e.  _V  ->  ( ~P (/)  C_  ~P B  ->  ~P (/)  ~<_  ~P B
) )
2318, 21, 22mpisyl 19 . 2  |-  ( A  ~<_  B  ->  ~P (/)  ~<_  ~P B
)
242, 15, 23pm2.61ne 2718 1  |-  ( A  ~<_  B  ->  ~P A  ~<_  ~P B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405   E.wex 1633    e. wcel 1842    =/= wne 2598   _Vcvv 3058    C_ wss 3413   (/)c0 3737   ~Pcpw 3954   class class class wbr 4394   -onto->wfo 5523    ~<_ cdom 7472    ~< csdm 7473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-er 7268  df-en 7475  df-dom 7476  df-sdom 7477
This theorem is referenced by:  cdalepw  8528  gchpwdom  8998  gchaclem  9006  2ndcredom  20135
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