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Theorem eqintg 25064
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, 15-Oct-2012.)
Hypotheses
Ref Expression
eqintg.1  |-  ( x  =  A  ->  ( ch 
<->  ps ) )
eqintg.2  |-  ( ph  ->  ch )
eqintg.3  |-  ( (
ph  /\  ps )  ->  A  C_  x )
Assertion
Ref Expression
eqintg  |-  ( (
ph  /\  A  e.  V )  ->  A  =  |^| { x  |  ps } )
Distinct variable groups:    x, A    ch, x    ph, x
Allowed substitution hints:    ps( x)    V( x)

Proof of Theorem eqintg
StepHypRef Expression
1 eqintg.3 . . . . . 6  |-  ( (
ph  /\  ps )  ->  A  C_  x )
21ex 423 . . . . 5  |-  ( ph  ->  ( ps  ->  A  C_  x ) )
32alrimiv 1621 . . . 4  |-  ( ph  ->  A. x ( ps 
->  A  C_  x ) )
43adantr 451 . . 3  |-  ( (
ph  /\  A  e.  V )  ->  A. x
( ps  ->  A  C_  x ) )
5 ssintab 3895 . . 3  |-  ( A 
C_  |^| { x  |  ps }  <->  A. x
( ps  ->  A  C_  x ) )
64, 5sylibr 203 . 2  |-  ( (
ph  /\  A  e.  V )  ->  A  C_ 
|^| { x  |  ps } )
7 eqintg.2 . . . . 5  |-  ( ph  ->  ch )
87adantr 451 . . . 4  |-  ( (
ph  /\  A  e.  V )  ->  ch )
9 simpr 447 . . . . 5  |-  ( (
ph  /\  A  e.  V )  ->  A  e.  V )
10 eqintg.1 . . . . . . 7  |-  ( x  =  A  ->  ( ch 
<->  ps ) )
1110bicomd 192 . . . . . 6  |-  ( x  =  A  ->  ( ps 
<->  ch ) )
1211ax-gen 1536 . . . . 5  |-  A. x
( x  =  A  ->  ( ps  <->  ch )
)
13 elabgt 2924 . . . . 5  |-  ( ( A  e.  V  /\  A. x ( x  =  A  ->  ( ps  <->  ch ) ) )  -> 
( A  e.  {
x  |  ps }  <->  ch ) )
149, 12, 13sylancl 643 . . . 4  |-  ( (
ph  /\  A  e.  V )  ->  ( A  e.  { x  |  ps }  <->  ch )
)
158, 14mpbird 223 . . 3  |-  ( (
ph  /\  A  e.  V )  ->  A  e.  { x  |  ps } )
16 intss1 3893 . . 3  |-  ( A  e.  { x  |  ps }  ->  |^| { x  |  ps }  C_  A
)
1715, 16syl 15 . 2  |-  ( (
ph  /\  A  e.  V )  ->  |^| { x  |  ps }  C_  A
)
186, 17eqssd 3209 1  |-  ( (
ph  /\  A  e.  V )  ->  A  =  |^| { x  |  ps } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530    = wceq 1632    e. wcel 1696   {cab 2282    C_ wss 3165   |^|cint 3878
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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