Theorem fbncp 24213

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) ∧ 𝐴 ∈ 𝐹) → ¬ (𝐵 ∖ 𝐴) ∈ 𝐹)

Proof of Theorem fbncp

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0nelfb 24205	. . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐹)
2	1	adantr 486	. 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹) → ¬ ∅ ∈ 𝐹)
3		fbasssin 24210	. . . 4 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ (𝐵 ∖ 𝐴) ∈ 𝐹) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ (𝐵 ∖ 𝐴)))
4		disjdif 4429	. . . . . . . 8 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅
5	4	sseq2i 3942	. . . . . . 7 ⊢ (𝑥 ⊆ (𝐴 ∩ (𝐵 ∖ 𝐴)) ↔ 𝑥 ⊆ ∅)
6		ss0 4355	. . . . . . 7 ⊢ (𝑥 ⊆ ∅ → 𝑥 = ∅)
7	5, 6	sylbi 220	. . . . . 6 ⊢ (𝑥 ⊆ (𝐴 ∩ (𝐵 ∖ 𝐴)) → 𝑥 = ∅)
8		eleq1 2852	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝐹 ↔ ∅ ∈ 𝐹))
9	8	biimpac 484	. . . . . 6 ⊢ ((𝑥 ∈ 𝐹 ∧ 𝑥 = ∅) → ∅ ∈ 𝐹)
10	7, 9	sylan2 605	. . . . 5 ⊢ ((𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ (𝐴 ∩ (𝐵 ∖ 𝐴))) → ∅ ∈ 𝐹)
11	10	rexlimiva 3159	. . . 4 ⊢ (∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ (𝐵 ∖ 𝐴)) → ∅ ∈ 𝐹)
12	3, 11	syl 18	. . 3 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ (𝐵 ∖ 𝐴) ∈ 𝐹) → ∅ ∈ 𝐹)
13	12	3expia 1139	. 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹) → ((𝐵 ∖ 𝐴) ∈ 𝐹 → ∅ ∈ 𝐹))
14	2, 13	mtod 201	1 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹) → ¬ (𝐵 ∖ 𝐴) ∈ 𝐹)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∃wrex 3090   ⊆ wss 3900   ∖ cdif 4048   ∩ cin 4092  ∅c0 4281  ‘cfv 6566  fBascfbas 21717

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fv 6574  df-fbas 21726

This theorem is used by:  filconn  24257  fgtr  24264  ufilb  24280  ufilmax  24281  ufilen  24304  flimrest  24357  fclsrest  24398  cfilres  25672  relcmpcmet  25694