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Theorem pwssb 3972
Description: Two ways to express a collection of subclasses. (Contributed by NM, 19-Jul-2006.)
Assertion
Ref Expression
pwssb  |-  ( A 
C_  ~P B  <->  A. x  e.  A  x  C_  B
)
Distinct variable groups:    x, A    x, B

Proof of Theorem pwssb
StepHypRef Expression
1 sspwuni 3971 . 2  |-  ( A 
C_  ~P B  <->  U. A  C_  B )
2 unissb 3839 . 2  |-  ( U. A  C_  B  <->  A. x  e.  A  x  C_  B
)
31, 2bitri 184 1  |-  ( A 
C_  ~P B  <->  A. x  e.  A  x  C_  B
)
Colors of variables: wff set class
Syntax hints:    <-> wb 105   A.wral 2455    C_ wss 3129   ~Pcpw 3575   U.cuni 3809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-in 3135  df-ss 3142  df-pw 3577  df-uni 3810
This theorem is referenced by: (None)
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