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Theorem sspwb 4034
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 3030 . . . . 5  |-  ( x 
C_  A  ->  ( A  C_  B  ->  x  C_  B ) )
21com12 30 . . . 4  |-  ( A 
C_  B  ->  (
x  C_  A  ->  x 
C_  B ) )
3 vex 2622 . . . . 5  |-  x  e. 
_V
43elpw 3431 . . . 4  |-  ( x  e.  ~P A  <->  x  C_  A
)
53elpw 3431 . . . 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 3029 . 2  |-  ( A 
C_  B  ->  ~P A  C_  ~P B )
8 ssel 3017 . . . 4  |-  ( ~P A  C_  ~P B  ->  ( { x }  e.  ~P A  ->  { x }  e.  ~P B
) )
93snex 4011 . . . . . 6  |-  { x }  e.  _V
109elpw 3431 . . . . 5  |-  ( { x }  e.  ~P A 
<->  { x }  C_  A )
113snss 3561 . . . . 5  |-  ( x  e.  A  <->  { x }  C_  A )
1210, 11bitr4i 185 . . . 4  |-  ( { x }  e.  ~P A 
<->  x  e.  A )
139elpw 3431 . . . . 5  |-  ( { x }  e.  ~P B 
<->  { x }  C_  B )
143snss 3561 . . . . 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 3029 . 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 1438    C_ wss 2997   ~Pcpw 3425   {csn 3441
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447
This theorem is referenced by:  pwel  4036  ssextss  4038  pweqb  4041
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