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Theorem isfne4b 26350
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 448 . . . . . . 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 2491 . . . . . 6  |-  ( ( B  e.  V  /\  X  =  Y )  ->  U. A  =  U. B )
5 uniexg 4706 . . . . . . 7  |-  ( B  e.  V  ->  U. B  e.  _V )
65adantr 452 . . . . . 6  |-  ( ( B  e.  V  /\  X  =  Y )  ->  U. B  e.  _V )
74, 6eqeltrd 2510 . . . . 5  |-  ( ( B  e.  V  /\  X  =  Y )  ->  U. A  e.  _V )
8 uniexb 4752 . . . . 5  |-  ( A  e.  _V  <->  U. A  e. 
_V )
97, 8sylibr 204 . . . 4  |-  ( ( B  e.  V  /\  X  =  Y )  ->  A  e.  _V )
10 simpl 444 . . . 4  |-  ( ( B  e.  V  /\  X  =  Y )  ->  B  e.  V )
11 tgss3 17051 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  V )  ->  ( ( topGen `  A
)  C_  ( topGen `  B )  <->  A  C_  ( topGen `
 B ) ) )
129, 10, 11syl2anc 643 . . 3  |-  ( ( B  e.  V  /\  X  =  Y )  ->  ( ( topGen `  A
)  C_  ( topGen `  B )  <->  A  C_  ( topGen `
 B ) ) )
1312pm5.32da 623 . 2  |-  ( B  e.  V  ->  (
( X  =  Y  /\  ( topGen `  A
)  C_  ( topGen `  B ) )  <->  ( X  =  Y  /\  A  C_  ( topGen `  B )
) ) )
142, 3isfne4 26349 . 2  |-  ( A Fne B  <->  ( X  =  Y  /\  A  C_  ( topGen `  B )
) )
1513, 14syl6rbbr 256 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 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:  fnetr  26366  fneval  26367
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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