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Theorem sspwb 3979
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 3007 . . . . 5  |-  ( x 
C_  A  ->  ( A  C_  B  ->  x  C_  B ) )
21com12 30 . . . 4  |-  ( A 
C_  B  ->  (
x  C_  A  ->  x 
C_  B ) )
3 vex 2605 . . . . 5  |-  x  e. 
_V
43elpw 3396 . . . 4  |-  ( x  e.  ~P A  <->  x  C_  A
)
53elpw 3396 . . . 4  |-  ( x  e.  ~P B  <->  x  C_  B
)
62, 4, 53imtr4g 203 . . 3  |-  ( A 
C_  B  ->  (
x  e.  ~P A  ->  x  e.  ~P B
) )
76ssrdv 3006 . 2  |-  ( A 
C_  B  ->  ~P A  C_  ~P B )
8 ssel 2994 . . . 4  |-  ( ~P A  C_  ~P B  ->  ( { x }  e.  ~P A  ->  { x }  e.  ~P B
) )
93snex 3965 . . . . . 6  |-  { x }  e.  _V
109elpw 3396 . . . . 5  |-  ( { x }  e.  ~P A 
<->  { x }  C_  A )
113snss 3524 . . . . 5  |-  ( x  e.  A  <->  { x }  C_  A )
1210, 11bitr4i 185 . . . 4  |-  ( { x }  e.  ~P A 
<->  x  e.  A )
139elpw 3396 . . . . 5  |-  ( { x }  e.  ~P B 
<->  { x }  C_  B )
143snss 3524 . . . . 5  |-  ( x  e.  B  <->  { x }  C_  B )
1513, 14bitr4i 185 . . . 4  |-  ( { x }  e.  ~P B 
<->  x  e.  B )
168, 12, 153imtr3g 202 . . 3  |-  ( ~P A  C_  ~P B  ->  ( x  e.  A  ->  x  e.  B ) )
1716ssrdv 3006 . 2  |-  ( ~P A  C_  ~P B  ->  A  C_  B )
187, 17impbii 124 1  |-  ( A 
C_  B  <->  ~P A  C_ 
~P B )
Colors of variables: wff set class
Syntax hints:    <-> wb 103    e. wcel 1434    C_ wss 2974   ~Pcpw 3390   {csn 3406
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412
This theorem is referenced by:  pwel  3981  ssextss  3983  pweqb  3986
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