Theorem ssmin 3842

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 3840	. 2 |- ( A C_ |^| { x | ( A C_ x /\ ph ) } <-> A. x ( ( A C_ x /\ ph ) -> A C_ x ) )
2		simpl 108	. 2 |- ( ( A C_ x /\ ph ) -> A C_ x )
3	1, 2	mpgbir 1441	1 |- A C_ |^| { x | ( A C_ x /\ ph ) }

Syntax hints:   -> wi 4   /\ wa 103   {cab 2151   C_ wss 3115   |^|cint 3823

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ral 2448  df-v 2727  df-in 3121  df-ss 3128  df-int 3824

This theorem is referenced by: (None)