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Theorem intmin3 3890
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 2915 . . 3  |-  ( A  e.  V  ->  ( A  e.  { x  |  ph }  <->  ps )
)
41, 3mpbiri 224 . 2  |-  ( A  e.  V  ->  A  e.  { x  |  ph } )
5 intss1 3877 . 2  |-  ( A  e.  { x  | 
ph }  ->  |^| { x  |  ph }  C_  A
)
64, 5syl 15 1  |-  ( A  e.  V  ->  |^| { x  |  ph }  C_  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   {cab 2269    C_ wss 3152   |^|cint 3862
This theorem is referenced by:  intabs  4172  intid  4231  eqint  24960  pfsubkl  26047
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-in 3159  df-ss 3166  df-int 3863
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