MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fbncp Structured version   Visualization version   GIF version

Theorem fbncp 21844
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 21836 . . 3 (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐹)
21adantr 472 . 2 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝐹) → ¬ ∅ ∈ 𝐹)
3 fbasssin 21841 . . . 4 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝐹 ∧ (𝐵𝐴) ∈ 𝐹) → ∃𝑥𝐹 𝑥 ⊆ (𝐴 ∩ (𝐵𝐴)))
4 disjdif 4184 . . . . . . . 8 (𝐴 ∩ (𝐵𝐴)) = ∅
54sseq2i 3771 . . . . . . 7 (𝑥 ⊆ (𝐴 ∩ (𝐵𝐴)) ↔ 𝑥 ⊆ ∅)
6 ss0 4117 . . . . . . 7 (𝑥 ⊆ ∅ → 𝑥 = ∅)
75, 6sylbi 207 . . . . . 6 (𝑥 ⊆ (𝐴 ∩ (𝐵𝐴)) → 𝑥 = ∅)
8 eleq1 2827 . . . . . . 7 (𝑥 = ∅ → (𝑥𝐹 ↔ ∅ ∈ 𝐹))
98biimpac 504 . . . . . 6 ((𝑥𝐹𝑥 = ∅) → ∅ ∈ 𝐹)
107, 9sylan2 492 . . . . 5 ((𝑥𝐹𝑥 ⊆ (𝐴 ∩ (𝐵𝐴))) → ∅ ∈ 𝐹)
1110rexlimiva 3166 . . . 4 (∃𝑥𝐹 𝑥 ⊆ (𝐴 ∩ (𝐵𝐴)) → ∅ ∈ 𝐹)
123, 11syl 17 . . 3 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝐹 ∧ (𝐵𝐴) ∈ 𝐹) → ∅ ∈ 𝐹)
13123expia 1115 . 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 383  w3a 1072   = wceq 1632  wcel 2139  wrex 3051  cdif 3712  cin 3714  wss 3715  c0 4058  cfv 6049  fBascfbas 19936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fv 6057  df-fbas 19945
This theorem is referenced by:  filconn  21888  fgtr  21895  ufilb  21911  ufilmax  21912  ufilen  21935  flimrest  21988  fclsrest  22029  cfilres  23294  relcmpcmet  23315
  Copyright terms: Public domain W3C validator