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Theorem fneint 30563
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 2475 . . . . 5  |-  ( x  =  y  ->  ( P  e.  x  <->  P  e.  y ) )
21elrab 3206 . . . 4  |-  ( y  e.  { x  e.  A  |  P  e.  x }  <->  ( y  e.  A  /\  P  e.  y ) )
3 fnessex 30561 . . . . . . 7  |-  ( ( A Fne B  /\  y  e.  A  /\  P  e.  y )  ->  E. z  e.  B  ( P  e.  z  /\  z  C_  y ) )
433expb 1198 . . . . . 6  |-  ( ( A Fne B  /\  ( y  e.  A  /\  P  e.  y
) )  ->  E. z  e.  B  ( P  e.  z  /\  z  C_  y ) )
5 eleq2 2475 . . . . . . . . . 10  |-  ( x  =  z  ->  ( P  e.  x  <->  P  e.  z ) )
65intminss 4253 . . . . . . . . 9  |-  ( ( z  e.  B  /\  P  e.  z )  ->  |^| { x  e.  B  |  P  e.  x }  C_  z
)
7 sstr 3449 . . . . . . . . 9  |-  ( (
|^| { x  e.  B  |  P  e.  x }  C_  z  /\  z  C_  y )  ->  |^| { x  e.  B  |  P  e.  x }  C_  y
)
86, 7sylan 469 . . . . . . . 8  |-  ( ( ( z  e.  B  /\  P  e.  z
)  /\  z  C_  y )  ->  |^| { x  e.  B  |  P  e.  x }  C_  y
)
98expl 616 . . . . . . 7  |-  ( z  e.  B  ->  (
( P  e.  z  /\  z  C_  y
)  ->  |^| { x  e.  B  |  P  e.  x }  C_  y
) )
109rexlimiv 2889 . . . . . 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 432 . . . 4  |-  ( A Fne B  ->  (
( y  e.  A  /\  P  e.  y
)  ->  |^| { x  e.  B  |  P  e.  x }  C_  y
) )
132, 12syl5bi 217 . . 3  |-  ( A Fne B  ->  (
y  e.  { x  e.  A  |  P  e.  x }  ->  |^| { x  e.  B  |  P  e.  x }  C_  y
) )
1413ralrimiv 2815 . 2  |-  ( A Fne B  ->  A. y  e.  { x  e.  A  |  P  e.  x } |^| { x  e.  B  |  P  e.  x }  C_  y
)
15 ssint 4242 . 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 212 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 367    e. wcel 1842   A.wral 2753   E.wrex 2754   {crab 2757    C_ wss 3413   |^|cint 4226   class class class wbr 4394   Fnecfne 30551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-int 4227  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-iota 5532  df-fun 5570  df-fv 5576  df-topgen 15056  df-fne 30552
This theorem is referenced by: (None)
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