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Theorem filinn0 23011
Description: The intersection of two elements of a filter can't be empty. (Contributed by FL, 16-Sep-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
Assertion
Ref Expression
filinn0 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → (𝐴𝐵) ≠ ∅)

Proof of Theorem filinn0
StepHypRef Expression
1 simp1 1135 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → 𝐹 ∈ (Fil‘𝑋))
2 filin 23005 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → (𝐴𝐵) ∈ 𝐹)
3 fileln0 23001 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴𝐵) ∈ 𝐹) → (𝐴𝐵) ≠ ∅)
41, 2, 3syl2anc 584 1 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → (𝐴𝐵) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086  wcel 2106  wne 2943  cin 3886  c0 4256  cfv 6433  Filcfil 22996
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
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-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  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-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fv 6441  df-fbas 20594  df-fil 22997
This theorem is referenced by:  flimclsi  23129  hausflimlem  23130  filnetlem3  34569
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