Theorem ssmin 3851

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 3849	. 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 1447	1 |- A C_ |^| { x | ( A C_ x /\ ph ) }

Syntax hints:   -> wi 4   /\ wa 103   {cab 2157   C_ wss 3122   |^|cint 3832

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 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-ext 2153

This theorem depends on definitions:  df-bi 116  df-tru 1352  df-nf 1455  df-sb 1757  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ral 2454  df-v 2733  df-in 3128  df-ss 3135  df-int 3833

This theorem is referenced by: (None)