Theorem ssmin 3893

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 3891	. 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 1467	1 |- A C_ |^| { x | ( A C_ x /\ ph ) }

Syntax hints:   -> wi 4   /\ wa 104   {cab 2182   C_ wss 3157   |^|cint 3874

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-in 3163  df-ss 3170  df-int 3875

This theorem is referenced by: (None)