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Theorem fneint 26247
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 2465 . . . . 5  |-  ( x  =  y  ->  ( P  e.  x  <->  P  e.  y ) )
21elrab 3052 . . . 4  |-  ( y  e.  { x  e.  A  |  P  e.  x }  <->  ( y  e.  A  /\  P  e.  y ) )
3 fnessex 26245 . . . . . . 7  |-  ( ( A Fne B  /\  y  e.  A  /\  P  e.  y )  ->  E. z  e.  B  ( P  e.  z  /\  z  C_  y ) )
433expb 1154 . . . . . 6  |-  ( ( A Fne B  /\  ( y  e.  A  /\  P  e.  y
) )  ->  E. z  e.  B  ( P  e.  z  /\  z  C_  y ) )
5 eleq2 2465 . . . . . . . . . 10  |-  ( x  =  z  ->  ( P  e.  x  <->  P  e.  z ) )
65intminss 4036 . . . . . . . . 9  |-  ( ( z  e.  B  /\  P  e.  z )  ->  |^| { x  e.  B  |  P  e.  x }  C_  z
)
7 sstr 3316 . . . . . . . . 9  |-  ( (
|^| { x  e.  B  |  P  e.  x }  C_  z  /\  z  C_  y )  ->  |^| { x  e.  B  |  P  e.  x }  C_  y
)
86, 7sylan 458 . . . . . . . 8  |-  ( ( ( z  e.  B  /\  P  e.  z
)  /\  z  C_  y )  ->  |^| { x  e.  B  |  P  e.  x }  C_  y
)
98expl 602 . . . . . . 7  |-  ( z  e.  B  ->  (
( P  e.  z  /\  z  C_  y
)  ->  |^| { x  e.  B  |  P  e.  x }  C_  y
) )
109rexlimiv 2784 . . . . . 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 424 . . . 4  |-  ( A Fne B  ->  (
( y  e.  A  /\  P  e.  y
)  ->  |^| { x  e.  B  |  P  e.  x }  C_  y
) )
132, 12syl5bi 209 . . 3  |-  ( A Fne B  ->  (
y  e.  { x  e.  A  |  P  e.  x }  ->  |^| { x  e.  B  |  P  e.  x }  C_  y
) )
1413ralrimiv 2748 . 2  |-  ( A Fne B  ->  A. y  e.  { x  e.  A  |  P  e.  x } |^| { x  e.  B  |  P  e.  x }  C_  y
)
15 ssint 4026 . 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 204 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 359    e. wcel 1721   A.wral 2666   E.wrex 2667   {crab 2670    C_ wss 3280   |^|cint 4010   class class class wbr 4172   Fnecfne 26229
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-int 4011  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-topgen 13622  df-fne 26233
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