Theorem intminss 3857

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 2887	. 2 |- ( A e. { x e. B | ph } <-> ( A e. B /\ ps ) )
3		intss1 3847	. 2 |- ( A e. { x e. B | ph } -> |^| { x e. B | ph } C_ A )
4	2, 3	sylbir 134	1 |- ( ( A e. B /\ ps ) -> |^| { x e. B | ph } C_ A )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1349   e. wcel 2142   {crab 2453   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-rab 2458  df-v 2733  df-in 3128  df-ss 3135  df-int 3833

This theorem is referenced by:  onintss  4376  cardonle  7168