Theorem fbncp 23702

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 23694	. . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐹)
2	1	adantr 480	. 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹) → ¬ ∅ ∈ 𝐹)
3		fbasssin 23699	. . . 4 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ (𝐵 ∖ 𝐴) ∈ 𝐹) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ (𝐵 ∖ 𝐴)))
4		disjdif 4431	. . . . . . . 8 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅
5	4	sseq2i 3973	. . . . . . 7 ⊢ (𝑥 ⊆ (𝐴 ∩ (𝐵 ∖ 𝐴)) ↔ 𝑥 ⊆ ∅)
6		ss0 4361	. . . . . . 7 ⊢ (𝑥 ⊆ ∅ → 𝑥 = ∅)
7	5, 6	sylbi 217	. . . . . 6 ⊢ (𝑥 ⊆ (𝐴 ∩ (𝐵 ∖ 𝐴)) → 𝑥 = ∅)
8		eleq1 2816	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝐹 ↔ ∅ ∈ 𝐹))
9	8	biimpac 478	. . . . . 6 ⊢ ((𝑥 ∈ 𝐹 ∧ 𝑥 = ∅) → ∅ ∈ 𝐹)
10	7, 9	sylan2 593	. . . . 5 ⊢ ((𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ (𝐴 ∩ (𝐵 ∖ 𝐴))) → ∅ ∈ 𝐹)
11	10	rexlimiva 3126	. . . 4 ⊢ (∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ (𝐵 ∖ 𝐴)) → ∅ ∈ 𝐹)
12	3, 11	syl 17	. . 3 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ (𝐵 ∖ 𝐴) ∈ 𝐹) → ∅ ∈ 𝐹)
13	12	3expia 1121	. 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹) → ((𝐵 ∖ 𝐴) ∈ 𝐹 → ∅ ∈ 𝐹))
14	2, 13	mtod 198	1 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹) → ¬ (𝐵 ∖ 𝐴) ∈ 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∃wrex 3053   ∖ cdif 3908   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292  ‘cfv 6499  fBascfbas 21228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fv 6507  df-fbas 21237

This theorem is referenced by:  filconn  23746  fgtr  23753  ufilb  23769  ufilmax  23770  ufilen  23793  flimrest  23846  fclsrest  23887  cfilres  25172  relcmpcmet  25194