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Theorem ssmin 3681
Description: Subclass of the minimum value of class of supersets. (Contributed by NM, 10-Aug-2006.)
Assertion
Ref Expression
ssmin  |-  A  C_  |^|
{ x  |  ( A  C_  x  /\  ph ) }
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem ssmin
StepHypRef Expression
1 ssintab 3679 . 2  |-  ( A 
C_  |^| { x  |  ( A  C_  x  /\  ph ) }  <->  A. x
( ( A  C_  x  /\  ph )  ->  A  C_  x ) )
2 simpl 107 . 2  |-  ( ( A  C_  x  /\  ph )  ->  A  C_  x
)
31, 2mpgbir 1383 1  |-  A  C_  |^|
{ x  |  ( A  C_  x  /\  ph ) }
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   {cab 2069    C_ wss 2984   |^|cint 3662
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-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2614  df-in 2990  df-ss 2997  df-int 3663
This theorem is referenced by: (None)
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