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Theorem sspwb 4108
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 3074 . . . . 5  |-  ( x 
C_  A  ->  ( A  C_  B  ->  x  C_  B ) )
21com12 30 . . . 4  |-  ( A 
C_  B  ->  (
x  C_  A  ->  x 
C_  B ) )
3 vex 2663 . . . . 5  |-  x  e. 
_V
43elpw 3486 . . . 4  |-  ( x  e.  ~P A  <->  x  C_  A
)
53elpw 3486 . . . 4  |-  ( x  e.  ~P B  <->  x  C_  B
)
62, 4, 53imtr4g 204 . . 3  |-  ( A 
C_  B  ->  (
x  e.  ~P A  ->  x  e.  ~P B
) )
76ssrdv 3073 . 2  |-  ( A 
C_  B  ->  ~P A  C_  ~P B )
8 ssel 3061 . . . 4  |-  ( ~P A  C_  ~P B  ->  ( { x }  e.  ~P A  ->  { x }  e.  ~P B
) )
93snex 4079 . . . . . 6  |-  { x }  e.  _V
109elpw 3486 . . . . 5  |-  ( { x }  e.  ~P A 
<->  { x }  C_  A )
113snss 3619 . . . . 5  |-  ( x  e.  A  <->  { x }  C_  A )
1210, 11bitr4i 186 . . . 4  |-  ( { x }  e.  ~P A 
<->  x  e.  A )
139elpw 3486 . . . . 5  |-  ( { x }  e.  ~P B 
<->  { x }  C_  B )
143snss 3619 . . . . 5  |-  ( x  e.  B  <->  { x }  C_  B )
1513, 14bitr4i 186 . . . 4  |-  ( { x }  e.  ~P B 
<->  x  e.  B )
168, 12, 153imtr3g 203 . . 3  |-  ( ~P A  C_  ~P B  ->  ( x  e.  A  ->  x  e.  B ) )
1716ssrdv 3073 . 2  |-  ( ~P A  C_  ~P B  ->  A  C_  B )
187, 17impbii 125 1  |-  ( A 
C_  B  <->  ~P A  C_ 
~P B )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    e. wcel 1465    C_ wss 3041   ~Pcpw 3480   {csn 3497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503
This theorem is referenced by:  pwel  4110  ssextss  4112  pweqb  4115  fiss  6833  ntrss  12215
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