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Theorem intminss 4274
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 3207 . 2  |-  ( A  e.  { x  e.  B  |  ph }  <->  ( A  e.  B  /\  ps ) )
3 intss1 4262 . 2  |-  ( A  e.  { x  e.  B  |  ph }  ->  |^| { x  e.  B  |  ph }  C_  A )
42, 3sylbir 218 1  |-  ( ( A  e.  B  /\  ps )  ->  |^| { x  e.  B  |  ph }  C_  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1454    e. wcel 1897   {crab 2752    C_ wss 3415   |^|cint 4247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-rab 2757  df-v 3058  df-in 3422  df-ss 3429  df-int 4248
This theorem is referenced by:  onintss  5491  knatar  6272  cardonle  8416  coftr  8728  wuncss  9195  ist1-3  20413  sigagenss  29019  ldgenpisyslem1  29033  dynkin  29037  nodenselem5  30622  nobndlem6  30634  nobndlem8  30636  fneint  31052  igenmin  32341  pclclN  33500  dfrcl2  36310
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