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Theorem intminss 3681
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 2757 . 2  |-  ( A  e.  { x  e.  B  |  ph }  <->  ( A  e.  B  /\  ps ) )
3 intss1 3671 . 2  |-  ( A  e.  { x  e.  B  |  ph }  ->  |^| { x  e.  B  |  ph }  C_  A )
42, 3sylbir 133 1  |-  ( ( A  e.  B  /\  ps )  ->  |^| { x  e.  B  |  ph }  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   {crab 2357    C_ wss 2982   |^|cint 3656
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rab 2362  df-v 2612  df-in 2988  df-ss 2995  df-int 3657
This theorem is referenced by:  onintss  4173  cardonle  6585
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