Theorem intminss 4249

Description: Under subset ordering, the intersection of a restricted class abstraction is less than or equal to any of its members. (Contributed by NM, 7-Sep-2013.)

Hypothesis

Ref	Expression
intminss.1	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
intminss	|- ( ( A e. B /\ ps ) -> |^| { x e. B | ph } C_ A )

Distinct variable groups:   x, A   x, B   ps, x

Allowed substitution hint:   ph( x)

Proof of Theorem intminss

Step	Hyp	Ref	Expression
1		intminss.1	. . 3 |- ( x = A -> ( ph <-> ps ) )
2	1	elrab 3211	. 2 |- ( A e. { x e. B | ph } <-> ( A e. B /\ ps ) )
3		intss1 4238	. 2 |- ( A e. { x e. B | ph } -> |^| { x e. B | ph } C_ A )
4	2, 3	sylbir 213	1 |- ( ( A e. B /\ ps ) -> |^| { x e. B | ph } C_ A )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1370   e. wcel 1758   {crab 2797   C_ wss 3423   |^|cint 4223

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-rab 2802  df-v 3067  df-in 3430  df-ss 3437  df-int 4224

This theorem is referenced by:  onintss  4864  knatar  6144  cardonle  8225  coftr  8540  wuncss  9010  ist1-3  19066  sigagenss  26723  nodenselem5  27957  nobndlem6  27969  nobndlem8  27971  fneint  28684  igenmin  28999  pclclN  33838