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Theorem ssmin 3881
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 3879 . 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 1537 1  |-  A  C_  |^|
{ x  |  ( A  C_  x  /\  ph ) }
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   {cab 2269    C_ wss 3152   |^|cint 3862
This theorem is referenced by:  tcid  7424
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-v 2790  df-in 3159  df-ss 3166  df-int 3863
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