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Theorem isfne4b 26270
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 2338 . . . . . 6  |-  ( ( B  e.  V  /\  X  =  Y )  ->  U. A  =  U. B )
5 uniexg 4517 . . . . . . 7  |-  ( B  e.  V  ->  U. B  e.  _V )
65adantr 451 . . . . . 6  |-  ( ( B  e.  V  /\  X  =  Y )  ->  U. B  e.  _V )
74, 6eqeltrd 2357 . . . . 5  |-  ( ( B  e.  V  /\  X  =  Y )  ->  U. A  e.  _V )
8 uniexb 4563 . . . . 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 16724 . . . 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 26269 . 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 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   U.cuni 3827   class class class wbr 4023   ` cfv 5255   topGenctg 13342   Fnecfne 26259
This theorem is referenced by:  fnetr  26286  fneval  26287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-topgen 13344  df-fne 26263
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