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Theorem fneint 30997
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 2517 . . . . 5  |-  ( x  =  y  ->  ( P  e.  x  <->  P  e.  y ) )
21elrab 3195 . . . 4  |-  ( y  e.  { x  e.  A  |  P  e.  x }  <->  ( y  e.  A  /\  P  e.  y ) )
3 fnessex 30995 . . . . . . 7  |-  ( ( A Fne B  /\  y  e.  A  /\  P  e.  y )  ->  E. z  e.  B  ( P  e.  z  /\  z  C_  y ) )
433expb 1208 . . . . . 6  |-  ( ( A Fne B  /\  ( y  e.  A  /\  P  e.  y
) )  ->  E. z  e.  B  ( P  e.  z  /\  z  C_  y ) )
5 eleq2 2517 . . . . . . . . . 10  |-  ( x  =  z  ->  ( P  e.  x  <->  P  e.  z ) )
65intminss 4260 . . . . . . . . 9  |-  ( ( z  e.  B  /\  P  e.  z )  ->  |^| { x  e.  B  |  P  e.  x }  C_  z
)
7 sstr 3439 . . . . . . . . 9  |-  ( (
|^| { x  e.  B  |  P  e.  x }  C_  z  /\  z  C_  y )  ->  |^| { x  e.  B  |  P  e.  x }  C_  y
)
86, 7sylan 474 . . . . . . . 8  |-  ( ( ( z  e.  B  /\  P  e.  z
)  /\  z  C_  y )  ->  |^| { x  e.  B  |  P  e.  x }  C_  y
)
98expl 623 . . . . . . 7  |-  ( z  e.  B  ->  (
( P  e.  z  /\  z  C_  y
)  ->  |^| { x  e.  B  |  P  e.  x }  C_  y
) )
109rexlimiv 2872 . . . . . 6  |-  ( E. z  e.  B  ( P  e.  z  /\  z  C_  y )  ->  |^| { x  e.  B  |  P  e.  x }  C_  y )
114, 10syl 17 . . . . 5  |-  ( ( A Fne B  /\  ( y  e.  A  /\  P  e.  y
) )  ->  |^| { x  e.  B  |  P  e.  x }  C_  y
)
1211ex 436 . . . 4  |-  ( A Fne B  ->  (
( y  e.  A  /\  P  e.  y
)  ->  |^| { x  e.  B  |  P  e.  x }  C_  y
) )
132, 12syl5bi 221 . . 3  |-  ( A Fne B  ->  (
y  e.  { x  e.  A  |  P  e.  x }  ->  |^| { x  e.  B  |  P  e.  x }  C_  y
) )
1413ralrimiv 2799 . 2  |-  ( A Fne B  ->  A. y  e.  { x  e.  A  |  P  e.  x } |^| { x  e.  B  |  P  e.  x }  C_  y
)
15 ssint 4249 . 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 216 1  |-  ( A Fne B  ->  |^| { x  e.  B  |  P  e.  x }  C_  |^| { x  e.  A  |  P  e.  x } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    e. wcel 1886   A.wral 2736   E.wrex 2737   {crab 2740    C_ wss 3403   |^|cint 4233   class class class wbr 4401   Fnecfne 30985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-int 4234  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-iota 5545  df-fun 5583  df-fv 5589  df-topgen 15335  df-fne 30986
This theorem is referenced by: (None)
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