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Theorem fnetr 28698
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 2451 . . . 4  |-  U. A  =  U. A
2 eqid 2451 . . . 4  |-  U. B  =  U. B
31, 2fnebas 28685 . . 3  |-  ( A Fne B  ->  U. A  =  U. B )
4 eqid 2451 . . . 4  |-  U. C  =  U. C
52, 4fnebas 28685 . . 3  |-  ( B Fne C  ->  U. B  =  U. C )
63, 5sylan9eq 2512 . 2  |-  ( ( A Fne B  /\  B Fne C )  ->  U. A  =  U. C )
7 fnerel 28679 . . . . 5  |-  Rel  Fne
87brrelex2i 4980 . . . 4  |-  ( A Fne B  ->  B  e.  _V )
91, 2isfne4b 28682 . . . . 5  |-  ( B  e.  _V  ->  ( A Fne B  <->  ( U. A  =  U. B  /\  ( topGen `  A )  C_  ( topGen `  B )
) ) )
109simplbda 624 . . . 4  |-  ( ( B  e.  _V  /\  A Fne B )  -> 
( topGen `  A )  C_  ( topGen `  B )
)
118, 10mpancom 669 . . 3  |-  ( A Fne B  ->  ( topGen `
 A )  C_  ( topGen `  B )
)
127brrelex2i 4980 . . . 4  |-  ( B Fne C  ->  C  e.  _V )
132, 4isfne4b 28682 . . . . 5  |-  ( C  e.  _V  ->  ( B Fne C  <->  ( U. B  =  U. C  /\  ( topGen `  B )  C_  ( topGen `  C )
) ) )
1413simplbda 624 . . . 4  |-  ( ( C  e.  _V  /\  B Fne C )  -> 
( topGen `  B )  C_  ( topGen `  C )
)
1512, 14mpancom 669 . . 3  |-  ( B Fne C  ->  ( topGen `
 B )  C_  ( topGen `  C )
)
1611, 15sylan9ss 3469 . 2  |-  ( ( A Fne B  /\  B Fne C )  -> 
( topGen `  A )  C_  ( topGen `  C )
)
1712adantl 466 . . 3  |-  ( ( A Fne B  /\  B Fne C )  ->  C  e.  _V )
181, 4isfne4b 28682 . . 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 913 1  |-  ( ( A Fne B  /\  B Fne C )  ->  A Fne C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3070    C_ wss 3428   U.cuni 4191   class class class wbr 4392   ` cfv 5518   topGenctg 14480   Fnecfne 28671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-iota 5481  df-fun 5520  df-fv 5526  df-topgen 14486  df-fne 28675
This theorem is referenced by:  fnessref  28705  fnemeet2  28728  fnejoin2  28730
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