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Theorem intmin3 3710
Description: Under subset ordering, the intersection of a class abstraction is less than or equal to any of its members. (Contributed by NM, 3-Jul-2005.)
Hypotheses
Ref Expression
intmin3.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
intmin3.3  |-  ps
Assertion
Ref Expression
intmin3  |-  ( A  e.  V  ->  |^| { x  |  ph }  C_  A
)
Distinct variable groups:    x, A    ps, x
Allowed substitution hints:    ph( x)    V( x)

Proof of Theorem intmin3
StepHypRef Expression
1 intmin3.3 . . 3  |-  ps
2 intmin3.2 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
32elabg 2759 . . 3  |-  ( A  e.  V  ->  ( A  e.  { x  |  ph }  <->  ps )
)
41, 3mpbiri 166 . 2  |-  ( A  e.  V  ->  A  e.  { x  |  ph } )
5 intss1 3698 . 2  |-  ( A  e.  { x  | 
ph }  ->  |^| { x  |  ph }  C_  A
)
64, 5syl 14 1  |-  ( A  e.  V  ->  |^| { x  |  ph }  C_  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1289    e. wcel 1438   {cab 2074    C_ wss 2997   |^|cint 3683
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3003  df-ss 3010  df-int 3684
This theorem is referenced by:  intid  4042
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