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Theorem eqint 25063
Description: To prove that a set  A is the finest one that has the property  ph, prove that  A is a part of all sets that has this property and that  A has also that property. (Contributed by FL, 21-Apr-2012.)
Hypotheses
Ref Expression
eqint.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
eqint.2  |-  ps
eqint.3  |-  ( ph  ->  A  C_  x )
Assertion
Ref Expression
eqint  |-  ( A  e.  V  ->  A  =  |^| { x  | 
ph } )
Distinct variable groups:    x, A    ps, x
Allowed substitution hints:    ph( x)    V( x)

Proof of Theorem eqint
StepHypRef Expression
1 ssintab 3895 . . . 4  |-  ( A 
C_  |^| { x  | 
ph }  <->  A. x
( ph  ->  A  C_  x ) )
2 eqint.3 . . . 4  |-  ( ph  ->  A  C_  x )
31, 2mpgbir 1540 . . 3  |-  A  C_  |^|
{ x  |  ph }
43a1i 10 . 2  |-  ( A  e.  V  ->  A  C_ 
|^| { x  |  ph } )
5 eqint.1 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
6 eqint.2 . . 3  |-  ps
75, 6intmin3 3906 . 2  |-  ( A  e.  V  ->  |^| { x  |  ph }  C_  A
)
84, 7eqssd 3209 1  |-  ( A  e.  V  ->  A  =  |^| { x  | 
ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696   {cab 2282    C_ wss 3165   |^|cint 3878
This theorem is referenced by:  intopcoaconb  25643
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-v 2803  df-in 3172  df-ss 3179  df-int 3879
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