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Theorem fnebas 33820
 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 33816 . 2 (𝐴Fne𝐵 ↔ (𝑋 = 𝑌𝐴 ⊆ (topGen‘𝐵)))
43simplbi 501 1 (𝐴Fne𝐵𝑋 = 𝑌)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ⊆ wss 3881  ∪ cuni 4801   class class class wbr 5031  ‘cfv 6325  topGenctg 16706  Fnecfne 33812 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-iota 6284  df-fun 6327  df-fv 6333  df-topgen 16712  df-fne 33813 This theorem is referenced by:  fnetr  33827  fnessref  33833  fnemeet2  33843  fnejoin2  33845
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