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Theorem fnetr 26389
Description: Transitivity of the fineness relation. (Contributed by Jeff Hankins, 5-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
Assertion
Ref Expression
fnetr  |-  ( ( A Fne B  /\  B Fne C )  ->  A Fne C )

Proof of Theorem fnetr
StepHypRef Expression
1 eqid 2296 . . . 4  |-  U. A  =  U. A
2 eqid 2296 . . . 4  |-  U. B  =  U. B
31, 2fnebas 26376 . . 3  |-  ( A Fne B  ->  U. A  =  U. B )
4 eqid 2296 . . . 4  |-  U. C  =  U. C
52, 4fnebas 26376 . . 3  |-  ( B Fne C  ->  U. B  =  U. C )
63, 5sylan9eq 2348 . 2  |-  ( ( A Fne B  /\  B Fne C )  ->  U. A  =  U. C )
7 fnerel 26370 . . . . 5  |-  Rel  Fne
87brrelex2i 4746 . . . 4  |-  ( A Fne B  ->  B  e.  _V )
91, 2isfne4b 26373 . . . . 5  |-  ( B  e.  _V  ->  ( A Fne B  <->  ( U. A  =  U. B  /\  ( topGen `  A )  C_  ( topGen `  B )
) ) )
109simplbda 607 . . . 4  |-  ( ( B  e.  _V  /\  A Fne B )  -> 
( topGen `  A )  C_  ( topGen `  B )
)
118, 10mpancom 650 . . 3  |-  ( A Fne B  ->  ( topGen `
 A )  C_  ( topGen `  B )
)
127brrelex2i 4746 . . . 4  |-  ( B Fne C  ->  C  e.  _V )
132, 4isfne4b 26373 . . . . 5  |-  ( C  e.  _V  ->  ( B Fne C  <->  ( U. B  =  U. C  /\  ( topGen `  B )  C_  ( topGen `  C )
) ) )
1413simplbda 607 . . . 4  |-  ( ( C  e.  _V  /\  B Fne C )  -> 
( topGen `  B )  C_  ( topGen `  C )
)
1512, 14mpancom 650 . . 3  |-  ( B Fne C  ->  ( topGen `
 B )  C_  ( topGen `  C )
)
1611, 15sylan9ss 3205 . 2  |-  ( ( A Fne B  /\  B Fne C )  -> 
( topGen `  A )  C_  ( topGen `  C )
)
1712adantl 452 . . 3  |-  ( ( A Fne B  /\  B Fne C )  ->  C  e.  _V )
181, 4isfne4b 26373 . . 3  |-  ( C  e.  _V  ->  ( A Fne C  <->  ( U. A  =  U. C  /\  ( topGen `  A )  C_  ( topGen `  C )
) ) )
1917, 18syl 15 . 2  |-  ( ( A Fne B  /\  B Fne C )  -> 
( A Fne C  <->  ( U. A  =  U. C  /\  ( topGen `  A
)  C_  ( topGen `  C ) ) ) )
206, 16, 19mpbir2and 888 1  |-  ( ( A Fne B  /\  B Fne C )  ->  A Fne C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   U.cuni 3843   class class class wbr 4039   ` cfv 5271   topGenctg 13358   Fnecfne 26362
This theorem is referenced by:  fnessref  26396  fnemeet2  26419  fnejoin2  26421
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-topgen 13360  df-fne 26366
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