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Theorem fnebas 36310
Description: A finer cover covers the same set as the original. (Contributed by Jeff Hankins, 28-Sep-2009.)
Hypotheses
Ref Expression
fnebas.1 𝑋 = 𝐴
fnebas.2 𝑌 = 𝐵
Assertion
Ref Expression
fnebas (𝐴Fne𝐵𝑋 = 𝑌)

Proof of Theorem fnebas
StepHypRef Expression
1 fnebas.1 . . 3 𝑋 = 𝐴
2 fnebas.2 . . 3 𝑌 = 𝐵
31, 2isfne4 36306 . 2 (𝐴Fne𝐵 ↔ (𝑋 = 𝑌𝐴 ⊆ (topGen‘𝐵)))
43simplbi 497 1 (𝐴Fne𝐵𝑋 = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wss 3976   cuni 4931   class class class wbr 5166  cfv 6573  topGenctg 17497  Fnecfne 36302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-topgen 17503  df-fne 36303
This theorem is referenced by:  fnetr  36317  fnessref  36323  fnemeet2  36333  fnejoin2  36335
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