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Theorem fneint 26371
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 2499 . . . . 5  |-  ( x  =  y  ->  ( P  e.  x  <->  P  e.  y ) )
21elrab 3094 . . . 4  |-  ( y  e.  { x  e.  A  |  P  e.  x }  <->  ( y  e.  A  /\  P  e.  y ) )
3 fnessex 26369 . . . . . . 7  |-  ( ( A Fne B  /\  y  e.  A  /\  P  e.  y )  ->  E. z  e.  B  ( P  e.  z  /\  z  C_  y ) )
433expb 1155 . . . . . 6  |-  ( ( A Fne B  /\  ( y  e.  A  /\  P  e.  y
) )  ->  E. z  e.  B  ( P  e.  z  /\  z  C_  y ) )
5 eleq2 2499 . . . . . . . . . 10  |-  ( x  =  z  ->  ( P  e.  x  <->  P  e.  z ) )
65intminss 4078 . . . . . . . . 9  |-  ( ( z  e.  B  /\  P  e.  z )  ->  |^| { x  e.  B  |  P  e.  x }  C_  z
)
7 sstr 3358 . . . . . . . . 9  |-  ( (
|^| { x  e.  B  |  P  e.  x }  C_  z  /\  z  C_  y )  ->  |^| { x  e.  B  |  P  e.  x }  C_  y
)
86, 7sylan 459 . . . . . . . 8  |-  ( ( ( z  e.  B  /\  P  e.  z
)  /\  z  C_  y )  ->  |^| { x  e.  B  |  P  e.  x }  C_  y
)
98expl 603 . . . . . . 7  |-  ( z  e.  B  ->  (
( P  e.  z  /\  z  C_  y
)  ->  |^| { x  e.  B  |  P  e.  x }  C_  y
) )
109rexlimiv 2826 . . . . . 6  |-  ( E. z  e.  B  ( P  e.  z  /\  z  C_  y )  ->  |^| { x  e.  B  |  P  e.  x }  C_  y )
114, 10syl 16 . . . . 5  |-  ( ( A Fne B  /\  ( y  e.  A  /\  P  e.  y
) )  ->  |^| { x  e.  B  |  P  e.  x }  C_  y
)
1211ex 425 . . . 4  |-  ( A Fne B  ->  (
( y  e.  A  /\  P  e.  y
)  ->  |^| { x  e.  B  |  P  e.  x }  C_  y
) )
132, 12syl5bi 210 . . 3  |-  ( A Fne B  ->  (
y  e.  { x  e.  A  |  P  e.  x }  ->  |^| { x  e.  B  |  P  e.  x }  C_  y
) )
1413ralrimiv 2790 . 2  |-  ( A Fne B  ->  A. y  e.  { x  e.  A  |  P  e.  x } |^| { x  e.  B  |  P  e.  x }  C_  y
)
15 ssint 4068 . 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 205 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 360    e. wcel 1726   A.wral 2707   E.wrex 2708   {crab 2711    C_ wss 3322   |^|cint 4052   class class class wbr 4215   Fnecfne 26353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-topgen 13672  df-fne 26357
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