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Theorem isfne4b 26373
Description: A condition for a topology to be finer than another. (Contributed by Jeff Hankins, 28-Sep-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)
Hypotheses
Ref Expression
isfne.1  |-  X  = 
U. A
isfne.2  |-  Y  = 
U. B
Assertion
Ref Expression
isfne4b  |-  ( B  e.  V  ->  ( A Fne B  <->  ( X  =  Y  /\  ( topGen `
 A )  C_  ( topGen `  B )
) ) )

Proof of Theorem isfne4b
StepHypRef Expression
1 simpr 447 . . . . . . 7  |-  ( ( B  e.  V  /\  X  =  Y )  ->  X  =  Y )
2 isfne.1 . . . . . . 7  |-  X  = 
U. A
3 isfne.2 . . . . . . 7  |-  Y  = 
U. B
41, 2, 33eqtr3g 2351 . . . . . 6  |-  ( ( B  e.  V  /\  X  =  Y )  ->  U. A  =  U. B )
5 uniexg 4533 . . . . . . 7  |-  ( B  e.  V  ->  U. B  e.  _V )
65adantr 451 . . . . . 6  |-  ( ( B  e.  V  /\  X  =  Y )  ->  U. B  e.  _V )
74, 6eqeltrd 2370 . . . . 5  |-  ( ( B  e.  V  /\  X  =  Y )  ->  U. A  e.  _V )
8 uniexb 4579 . . . . 5  |-  ( A  e.  _V  <->  U. A  e. 
_V )
97, 8sylibr 203 . . . 4  |-  ( ( B  e.  V  /\  X  =  Y )  ->  A  e.  _V )
10 simpl 443 . . . 4  |-  ( ( B  e.  V  /\  X  =  Y )  ->  B  e.  V )
11 tgss3 16740 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  V )  ->  ( ( topGen `  A
)  C_  ( topGen `  B )  <->  A  C_  ( topGen `
 B ) ) )
129, 10, 11syl2anc 642 . . 3  |-  ( ( B  e.  V  /\  X  =  Y )  ->  ( ( topGen `  A
)  C_  ( topGen `  B )  <->  A  C_  ( topGen `
 B ) ) )
1312pm5.32da 622 . 2  |-  ( B  e.  V  ->  (
( X  =  Y  /\  ( topGen `  A
)  C_  ( topGen `  B ) )  <->  ( X  =  Y  /\  A  C_  ( topGen `  B )
) ) )
142, 3isfne4 26372 . 2  |-  ( A Fne B  <->  ( X  =  Y  /\  A  C_  ( topGen `  B )
) )
1513, 14syl6rbbr 255 1  |-  ( B  e.  V  ->  ( A Fne B  <->  ( X  =  Y  /\  ( topGen `
 A )  C_  ( topGen `  B )
) ) )
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:  fnetr  26389  fneval  26390
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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