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Theorem intmin3 3766
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 2801 . . 3  |-  ( A  e.  V  ->  ( A  e.  { x  |  ph }  <->  ps )
)
41, 3mpbiri 167 . 2  |-  ( A  e.  V  ->  A  e.  { x  |  ph } )
5 intss1 3754 . 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 104    = wceq 1314    e. wcel 1463   {cab 2101    C_ wss 3039   |^|cint 3739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-in 3045  df-ss 3052  df-int 3740
This theorem is referenced by:  intid  4114
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