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Theorem intmin3 4070
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 3075 . . 3  |-  ( A  e.  V  ->  ( A  e.  { x  |  ph }  <->  ps )
)
41, 3mpbiri 225 . 2  |-  ( A  e.  V  ->  A  e.  { x  |  ph } )
5 intss1 4057 . 2  |-  ( A  e.  { x  | 
ph }  ->  |^| { x  |  ph }  C_  A
)
64, 5syl 16 1  |-  ( A  e.  V  ->  |^| { x  |  ph }  C_  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725   {cab 2421    C_ wss 3312   |^|cint 4042
This theorem is referenced by:  intabs  4353  intid  4413
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-in 3319  df-ss 3326  df-int 4043
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