Theorem intminss 4298

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 3254	. 2 |- ( A e. { x e. B | ph } <-> ( A e. B /\ ps ) )
3		intss1 4286	. 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 367   = wceq 1398   e. wcel 1823   {crab 2808   C_ wss 3461   |^|cint 4271

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rab 2813  df-v 3108  df-in 3468  df-ss 3475  df-int 4272

This theorem is referenced by:  onintss  4917  knatar  6228  cardonle  8329  coftr  8644  wuncss  9112  ist1-3  20017  sigagenss  28379  nodenselem5  29685  nobndlem6  29697  nobndlem8  29699  fneint  30406  igenmin  30701  pclclN  36012  dfrcl2  38193