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Theorem fneint 25784
Description: If a cover is finer than another, every point can be approached more closely by intersections. (Contributed by Jeff Hankins, 11-Oct-2009.)
Assertion
Ref Expression
fneint  |-  ( A Fne B  ->  |^| { x  e.  B  |  P  e.  x }  C_  |^| { x  e.  A  |  P  e.  x } )
Distinct variable groups:    x, A    x, B    x, P

Proof of Theorem fneint
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2427 . . . . 5  |-  ( x  =  y  ->  ( P  e.  x  <->  P  e.  y ) )
21elrab 3009 . . . 4  |-  ( y  e.  { x  e.  A  |  P  e.  x }  <->  ( y  e.  A  /\  P  e.  y ) )
3 fnessex 25782 . . . . . . 7  |-  ( ( A Fne B  /\  y  e.  A  /\  P  e.  y )  ->  E. z  e.  B  ( P  e.  z  /\  z  C_  y ) )
433expb 1153 . . . . . 6  |-  ( ( A Fne B  /\  ( y  e.  A  /\  P  e.  y
) )  ->  E. z  e.  B  ( P  e.  z  /\  z  C_  y ) )
5 eleq2 2427 . . . . . . . . . 10  |-  ( x  =  z  ->  ( P  e.  x  <->  P  e.  z ) )
65intminss 3990 . . . . . . . . 9  |-  ( ( z  e.  B  /\  P  e.  z )  ->  |^| { x  e.  B  |  P  e.  x }  C_  z
)
7 sstr 3273 . . . . . . . . 9  |-  ( (
|^| { x  e.  B  |  P  e.  x }  C_  z  /\  z  C_  y )  ->  |^| { x  e.  B  |  P  e.  x }  C_  y
)
86, 7sylan 457 . . . . . . . 8  |-  ( ( ( z  e.  B  /\  P  e.  z
)  /\  z  C_  y )  ->  |^| { x  e.  B  |  P  e.  x }  C_  y
)
98expl 601 . . . . . . 7  |-  ( z  e.  B  ->  (
( P  e.  z  /\  z  C_  y
)  ->  |^| { x  e.  B  |  P  e.  x }  C_  y
) )
109rexlimiv 2746 . . . . . 6  |-  ( E. z  e.  B  ( P  e.  z  /\  z  C_  y )  ->  |^| { x  e.  B  |  P  e.  x }  C_  y )
114, 10syl 15 . . . . 5  |-  ( ( A Fne B  /\  ( y  e.  A  /\  P  e.  y
) )  ->  |^| { x  e.  B  |  P  e.  x }  C_  y
)
1211ex 423 . . . 4  |-  ( A Fne B  ->  (
( y  e.  A  /\  P  e.  y
)  ->  |^| { x  e.  B  |  P  e.  x }  C_  y
) )
132, 12syl5bi 208 . . 3  |-  ( A Fne B  ->  (
y  e.  { x  e.  A  |  P  e.  x }  ->  |^| { x  e.  B  |  P  e.  x }  C_  y
) )
1413ralrimiv 2710 . 2  |-  ( A Fne B  ->  A. y  e.  { x  e.  A  |  P  e.  x } |^| { x  e.  B  |  P  e.  x }  C_  y
)
15 ssint 3980 . 2  |-  ( |^| { x  e.  B  |  P  e.  x }  C_ 
|^| { x  e.  A  |  P  e.  x } 
<-> 
A. y  e.  {
x  e.  A  |  P  e.  x } |^| { x  e.  B  |  P  e.  x }  C_  y )
1614, 15sylibr 203 1  |-  ( A Fne B  ->  |^| { x  e.  B  |  P  e.  x }  C_  |^| { x  e.  A  |  P  e.  x } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1715   A.wral 2628   E.wrex 2629   {crab 2632    C_ wss 3238   |^|cint 3964   class class class wbr 4125   Fnecfne 25766
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-int 3965  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-iota 5322  df-fun 5360  df-fv 5366  df-topgen 13554  df-fne 25770
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