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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
StepHypRef Expression
1 intminss.1 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
21elrab 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 )
42, 3sylbir 205 1  |-  ( ( A  e.  B  /\  ps )  ->  |^| { x  e.  B  |  ph }  C_  A )
Colors of variables: wff set class
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
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