MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssmin Structured version   Unicode version

Theorem ssmin 4071
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 4069 . 2  |-  ( A 
C_  |^| { x  |  ( A  C_  x  /\  ph ) }  <->  A. x
( ( A  C_  x  /\  ph )  ->  A  C_  x ) )
2 simpl 445 . 2  |-  ( ( A  C_  x  /\  ph )  ->  A  C_  x
)
31, 2mpgbir 1560 1  |-  A  C_  |^|
{ x  |  ( A  C_  x  /\  ph ) }
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360   {cab 2424    C_ wss 3322   |^|cint 4052
This theorem is referenced by:  tcid  7681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-v 2960  df-in 3329  df-ss 3336  df-int 4053
  Copyright terms: Public domain W3C validator