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Theorem intminss 3888
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
StepHypRef Expression
1 intminss.1 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
21elrab 2923 . 2  |-  ( A  e.  { x  e.  B  |  ph }  <->  ( A  e.  B  /\  ps ) )
3 intss1 3877 . 2  |-  ( A  e.  { x  e.  B  |  ph }  ->  |^| { x  e.  B  |  ph }  C_  A )
42, 3sylbir 204 1  |-  ( ( A  e.  B  /\  ps )  ->  |^| { x  e.  B  |  ph }  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547    C_ wss 3152   |^|cint 3862
This theorem is referenced by:  onintss  4442  knatar  5857  cardonle  7590  coftr  7899  wuncss  8367  ist1-3  17077  nodenselem5  24339  nobndlem6  24351  nobndlem8  24353  fneint  26277  igenmin  26689  pclclN  30080
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-in 3159  df-ss 3166  df-int 3863
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