Theorem intminss 4036

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 3052	. 2 |- ( A e. { x e. B | ph } <-> ( A e. B /\ ps ) )
3		intss1 4025	. 2 |- ( A e. { x e. B | ph } -> |^| { x e. B | ph } C_ A )
4	2, 3	sylbir 205	1 |- ( ( A e. B /\ ps ) -> |^| { x e. B | ph } C_ A )

Syntax hints:   -> wi 4   <-> wb 177   /\ wa 359   = wceq 1649   e. wcel 1721   {crab 2670   C_ wss 3280   |^|cint 4010

This theorem is referenced by:  onintss  4591  knatar  6039  cardonle  7800  coftr  8109  wuncss  8576  ist1-3  17367  sigagenss  24485  nodenselem5  25553  nobndlem6  25565  nobndlem8  25567  fneint  26247  igenmin  26564  pclclN  30373

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385

This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rab 2675  df-v 2918  df-in 3287  df-ss 3294  df-int 4011