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Theorem fnetr 26366
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 2436 . . . 4  |-  U. A  =  U. A
2 eqid 2436 . . . 4  |-  U. B  =  U. B
31, 2fnebas 26353 . . 3  |-  ( A Fne B  ->  U. A  =  U. B )
4 eqid 2436 . . . 4  |-  U. C  =  U. C
52, 4fnebas 26353 . . 3  |-  ( B Fne C  ->  U. B  =  U. C )
63, 5sylan9eq 2488 . 2  |-  ( ( A Fne B  /\  B Fne C )  ->  U. A  =  U. C )
7 fnerel 26347 . . . . 5  |-  Rel  Fne
87brrelex2i 4919 . . . 4  |-  ( A Fne B  ->  B  e.  _V )
91, 2isfne4b 26350 . . . . 5  |-  ( B  e.  _V  ->  ( A Fne B  <->  ( U. A  =  U. B  /\  ( topGen `  A )  C_  ( topGen `  B )
) ) )
109simplbda 608 . . . 4  |-  ( ( B  e.  _V  /\  A Fne B )  -> 
( topGen `  A )  C_  ( topGen `  B )
)
118, 10mpancom 651 . . 3  |-  ( A Fne B  ->  ( topGen `
 A )  C_  ( topGen `  B )
)
127brrelex2i 4919 . . . 4  |-  ( B Fne C  ->  C  e.  _V )
132, 4isfne4b 26350 . . . . 5  |-  ( C  e.  _V  ->  ( B Fne C  <->  ( U. B  =  U. C  /\  ( topGen `  B )  C_  ( topGen `  C )
) ) )
1413simplbda 608 . . . 4  |-  ( ( C  e.  _V  /\  B Fne C )  -> 
( topGen `  B )  C_  ( topGen `  C )
)
1512, 14mpancom 651 . . 3  |-  ( B Fne C  ->  ( topGen `
 B )  C_  ( topGen `  C )
)
1611, 15sylan9ss 3361 . 2  |-  ( ( A Fne B  /\  B Fne C )  -> 
( topGen `  A )  C_  ( topGen `  C )
)
1712adantl 453 . . 3  |-  ( ( A Fne B  /\  B Fne C )  ->  C  e.  _V )
181, 4isfne4b 26350 . . 3  |-  ( C  e.  _V  ->  ( A Fne C  <->  ( U. A  =  U. C  /\  ( topGen `  A )  C_  ( topGen `  C )
) ) )
1917, 18syl 16 . 2  |-  ( ( A Fne B  /\  B Fne C )  -> 
( A Fne C  <->  ( U. A  =  U. C  /\  ( topGen `  A
)  C_  ( topGen `  C ) ) ) )
206, 16, 19mpbir2and 889 1  |-  ( ( A Fne B  /\  B Fne C )  ->  A Fne C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956    C_ wss 3320   U.cuni 4015   class class class wbr 4212   ` cfv 5454   topGenctg 13665   Fnecfne 26339
This theorem is referenced by:  fnessref  26373  fnemeet2  26396  fnejoin2  26398
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-topgen 13667  df-fne 26343
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