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Theorem fnebas 26044
Description: A finer cover covers the same set as the original. (Contributed by Jeff Hankins, 28-Sep-2009.)
Hypotheses
Ref Expression
fnebas.1  |-  X  = 
U. A
fnebas.2  |-  Y  = 
U. B
Assertion
Ref Expression
fnebas  |-  ( A Fne B  ->  X  =  Y )

Proof of Theorem fnebas
StepHypRef Expression
1 fnebas.1 . . 3  |-  X  = 
U. A
2 fnebas.2 . . 3  |-  Y  = 
U. B
31, 2isfne4 26040 . 2  |-  ( A Fne B  <->  ( X  =  Y  /\  A  C_  ( topGen `  B )
) )
43simplbi 447 1  |-  ( A Fne B  ->  X  =  Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    C_ wss 3263   U.cuni 3957   class class class wbr 4153   ` cfv 5394   topGenctg 13592   Fnecfne 26030
This theorem is referenced by:  fnetr  26057  fnessref  26064  fnemeet2  26087  fnejoin2  26089
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402  df-topgen 13594  df-fne 26034
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