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Theorem fbncp 21453
 Description: A filter base does not contain complements of its elements. (Contributed by Mario Carneiro, 26-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)
Assertion
Ref Expression
fbncp ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝐹) → ¬ (𝐵𝐴) ∈ 𝐹)

Proof of Theorem fbncp
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 0nelfb 21445 . . 3 (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐹)
21adantr 480 . 2 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝐹) → ¬ ∅ ∈ 𝐹)
3 fbasssin 21450 . . . 4 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝐹 ∧ (𝐵𝐴) ∈ 𝐹) → ∃𝑥𝐹 𝑥 ⊆ (𝐴 ∩ (𝐵𝐴)))
4 disjdif 3992 . . . . . . . 8 (𝐴 ∩ (𝐵𝐴)) = ∅
54sseq2i 3593 . . . . . . 7 (𝑥 ⊆ (𝐴 ∩ (𝐵𝐴)) ↔ 𝑥 ⊆ ∅)
6 ss0 3926 . . . . . . 7 (𝑥 ⊆ ∅ → 𝑥 = ∅)
75, 6sylbi 206 . . . . . 6 (𝑥 ⊆ (𝐴 ∩ (𝐵𝐴)) → 𝑥 = ∅)
8 eleq1 2676 . . . . . . 7 (𝑥 = ∅ → (𝑥𝐹 ↔ ∅ ∈ 𝐹))
98biimpac 502 . . . . . 6 ((𝑥𝐹𝑥 = ∅) → ∅ ∈ 𝐹)
107, 9sylan2 490 . . . . 5 ((𝑥𝐹𝑥 ⊆ (𝐴 ∩ (𝐵𝐴))) → ∅ ∈ 𝐹)
1110rexlimiva 3010 . . . 4 (∃𝑥𝐹 𝑥 ⊆ (𝐴 ∩ (𝐵𝐴)) → ∅ ∈ 𝐹)
123, 11syl 17 . . 3 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝐹 ∧ (𝐵𝐴) ∈ 𝐹) → ∅ ∈ 𝐹)
13123expia 1259 . 2 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝐹) → ((𝐵𝐴) ∈ 𝐹 → ∅ ∈ 𝐹))
142, 13mtod 188 1 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝐹) → ¬ (𝐵𝐴) ∈ 𝐹)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  ‘cfv 5804  fBascfbas 19555 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-fbas 19564 This theorem is referenced by:  filcon  21497  fgtr  21504  ufilb  21520  ufilmax  21521  ufilen  21544  flimrest  21597  fclsrest  21638  cfilres  22902  relcmpcmet  22923
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