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Theorem ssmin 3897
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 3895 . 2  |-  ( A 
C_  |^| { x  |  ( A  C_  x  /\  ph ) }  <->  A. x
( ( A  C_  x  /\  ph )  ->  A  C_  x ) )
2 simpl 443 . 2  |-  ( ( A  C_  x  /\  ph )  ->  A  C_  x
)
31, 2mpgbir 1540 1  |-  A  C_  |^|
{ x  |  ( A  C_  x  /\  ph ) }
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   {cab 2282    C_ wss 3165   |^|cint 3878
This theorem is referenced by:  tcid  7440
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-v 2803  df-in 3172  df-ss 3179  df-int 3879
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