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Theorem fbncp 21553
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 21545 . . 3 (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐹)
21adantr 481 . 2 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝐹) → ¬ ∅ ∈ 𝐹)
3 fbasssin 21550 . . . 4 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝐹 ∧ (𝐵𝐴) ∈ 𝐹) → ∃𝑥𝐹 𝑥 ⊆ (𝐴 ∩ (𝐵𝐴)))
4 disjdif 4012 . . . . . . . 8 (𝐴 ∩ (𝐵𝐴)) = ∅
54sseq2i 3609 . . . . . . 7 (𝑥 ⊆ (𝐴 ∩ (𝐵𝐴)) ↔ 𝑥 ⊆ ∅)
6 ss0 3946 . . . . . . 7 (𝑥 ⊆ ∅ → 𝑥 = ∅)
75, 6sylbi 207 . . . . . 6 (𝑥 ⊆ (𝐴 ∩ (𝐵𝐴)) → 𝑥 = ∅)
8 eleq1 2686 . . . . . . 7 (𝑥 = ∅ → (𝑥𝐹 ↔ ∅ ∈ 𝐹))
98biimpac 503 . . . . . 6 ((𝑥𝐹𝑥 = ∅) → ∅ ∈ 𝐹)
107, 9sylan2 491 . . . . 5 ((𝑥𝐹𝑥 ⊆ (𝐴 ∩ (𝐵𝐴))) → ∅ ∈ 𝐹)
1110rexlimiva 3021 . . . 4 (∃𝑥𝐹 𝑥 ⊆ (𝐴 ∩ (𝐵𝐴)) → ∅ ∈ 𝐹)
123, 11syl 17 . . 3 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝐹 ∧ (𝐵𝐴) ∈ 𝐹) → ∅ ∈ 𝐹)
13123expia 1264 . 2 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝐹) → ((𝐵𝐴) ∈ 𝐹 → ∅ ∈ 𝐹))
142, 13mtod 189 1 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝐹) → ¬ (𝐵𝐴) ∈ 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wrex 2908  cdif 3552  cin 3554  wss 3555  c0 3891  cfv 5847  fBascfbas 19653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fv 5855  df-fbas 19662
This theorem is referenced by:  filconn  21597  fgtr  21604  ufilb  21620  ufilmax  21621  ufilen  21644  flimrest  21697  fclsrest  21738  cfilres  23002  relcmpcmet  23023
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