Theorem ssmin 3790

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 3788	. 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 1429	1 |- A C_ |^| { x | ( A C_ x /\ ph ) }

Syntax hints:   -> wi 4   /\ wa 103   {cab 2125   C_ wss 3071   |^|cint 3771

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-in 3077  df-ss 3084  df-int 3772

This theorem is referenced by: (None)