Theorem fnebas 34533

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	⊢ (𝐴Fne𝐵 → 𝑋 = 𝑌)

Proof of Theorem fnebas

Step	Hyp	Ref	Expression
1		fnebas.1	. . 3 ⊢ 𝑋 = ∪ 𝐴
2		fnebas.2	. . 3 ⊢ 𝑌 = ∪ 𝐵
3	1, 2	isfne4 34529	. 2 ⊢ (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)))
4	3	simplbi 498	1 ⊢ (𝐴Fne𝐵 → 𝑋 = 𝑌)

Syntax hints:   → wi 4   = wceq 1539   ⊆ wss 3887  ∪ cuni 4839   class class class wbr 5074  ‘cfv 6433  topGenctg 17148  Fnecfne 34525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-topgen 17154  df-fne 34526

This theorem is referenced by:  fnetr  34540  fnessref  34546  fnemeet2  34556  fnejoin2  34558