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Theorem ssmin 4037
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 4035 . 2  |-  ( A 
C_  |^| { x  |  ( A  C_  x  /\  ph ) }  <->  A. x
( ( A  C_  x  /\  ph )  ->  A  C_  x ) )
2 simpl 444 . 2  |-  ( ( A  C_  x  /\  ph )  ->  A  C_  x
)
31, 2mpgbir 1556 1  |-  A  C_  |^|
{ x  |  ( A  C_  x  /\  ph ) }
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   {cab 2398    C_ wss 3288   |^|cint 4018
This theorem is referenced by:  tcid  7642
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ral 2679  df-v 2926  df-in 3295  df-ss 3302  df-int 4019
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