Theorem ssmin 3889

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

Step	Hyp	Ref	Expression
1		ssintab 3887	. 2 |- ( A C_ |^| { x | ( A C_ x /\ ph ) } <-> A. x ( ( A C_ x /\ ph ) -> A C_ x ) )
2		simpl 109	. 2 |- ( ( A C_ x /\ ph ) -> A C_ x )
3	1, 2	mpgbir 1464	1 |- A C_ |^| { x | ( A C_ x /\ ph ) }

Syntax hints:   -> wi 4   /\ wa 104   {cab 2179   C_ wss 3153   |^|cint 3870

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-in 3159  df-ss 3166  df-int 3871

This theorem is referenced by: (None)