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Theorem sspwb 4223
Description: Classes are subclasses if and only if their power classes are subclasses. Exercise 18 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.)
Assertion
Ref Expression
sspwb  |-  ( A 
C_  B  <->  ~P A  C_ 
~P B )

Proof of Theorem sspwb
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 sstr2 3186 . . . . 5  |-  ( x 
C_  A  ->  ( A  C_  B  ->  x  C_  B ) )
21com12 27 . . . 4  |-  ( A 
C_  B  ->  (
x  C_  A  ->  x 
C_  B ) )
3 vex 2791 . . . . 5  |-  x  e. 
_V
43elpw 3631 . . . 4  |-  ( x  e.  ~P A  <->  x  C_  A
)
53elpw 3631 . . . 4  |-  ( x  e.  ~P B  <->  x  C_  B
)
62, 4, 53imtr4g 261 . . 3  |-  ( A 
C_  B  ->  (
x  e.  ~P A  ->  x  e.  ~P B
) )
76ssrdv 3185 . 2  |-  ( A 
C_  B  ->  ~P A  C_  ~P B )
8 ssel 3174 . . . 4  |-  ( ~P A  C_  ~P B  ->  ( { x }  e.  ~P A  ->  { x }  e.  ~P B
) )
9 snex 4216 . . . . . 6  |-  { x }  e.  _V
109elpw 3631 . . . . 5  |-  ( { x }  e.  ~P A 
<->  { x }  C_  A )
113snss 3748 . . . . 5  |-  ( x  e.  A  <->  { x }  C_  A )
1210, 11bitr4i 243 . . . 4  |-  ( { x }  e.  ~P A 
<->  x  e.  A )
139elpw 3631 . . . . 5  |-  ( { x }  e.  ~P B 
<->  { x }  C_  B )
143snss 3748 . . . . 5  |-  ( x  e.  B  <->  { x }  C_  B )
1513, 14bitr4i 243 . . . 4  |-  ( { x }  e.  ~P B 
<->  x  e.  B )
168, 12, 153imtr3g 260 . . 3  |-  ( ~P A  C_  ~P B  ->  ( x  e.  A  ->  x  e.  B ) )
1716ssrdv 3185 . 2  |-  ( ~P A  C_  ~P B  ->  A  C_  B )
187, 17impbii 180 1  |-  ( A 
C_  B  <->  ~P A  C_ 
~P B )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    e. wcel 1684    C_ wss 3152   ~Pcpw 3625   {csn 3640
This theorem is referenced by:  pwel  4225  ssextss  4227  pweqb  4230  pwdom  7013  marypha1lem  7186  wdompwdom  7292  r1pwss  7456  pwwf  7479  rankpwi  7495  rankxplim  7549  ackbij2lem1  7845  fictb  7871  ssfin2  7946  ssfin3ds  7956  ttukeylem2  8137  hashbclem  11390  wrdexg  11425  incexclem  12295  hashbcss  13051  isacs1i  13559  mreacs  13560  acsfn  13561  sscpwex  13692  wunfunc  13773  isacs3lem  14269  isacs5lem  14272  tgcmp  17128  imastopn  17411  fgabs  17574  fgtr  17585  trfg  17586  ssufl  17613  alexsubb  17740  tsmsres  17826  cfilresi  18721  cmetss  18740  minveclem4a  18794  minveclem4  18796  vitali  18968  sqff1o  20420  ballotlem2  23047  elsigagen2  23509  measres  23549  imambfm  23567  lemindclsbu  25995  neibastop1  26308  neibastop2lem  26309  neibastop2  26310  sstotbnd2  26498  isnacs3  26785  aomclem2  27152
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-pw 3627  df-sn 3646  df-pr 3647
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