Theorem fbncp 23755

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 23747	. . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐹)
2	1	adantr 480	. 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹) → ¬ ∅ ∈ 𝐹)
3		fbasssin 23752	. . . 4 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ (𝐵 ∖ 𝐴) ∈ 𝐹) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ (𝐵 ∖ 𝐴)))
4		disjdif 4421	. . . . . . . 8 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅
5	4	sseq2i 3960	. . . . . . 7 ⊢ (𝑥 ⊆ (𝐴 ∩ (𝐵 ∖ 𝐴)) ↔ 𝑥 ⊆ ∅)
6		ss0 4351	. . . . . . 7 ⊢ (𝑥 ⊆ ∅ → 𝑥 = ∅)
7	5, 6	sylbi 217	. . . . . 6 ⊢ (𝑥 ⊆ (𝐴 ∩ (𝐵 ∖ 𝐴)) → 𝑥 = ∅)
8		eleq1 2821	. . . . . . 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 1541   ∈ wcel 2113  ∃wrex 3057   ∖ cdif 3895   ∩ cin 3897   ⊆ wss 3898  ∅c0 4282  ‘cfv 6486  fBascfbas 21281

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fv 6494  df-fbas 21290

This theorem is referenced by:  filconn  23799  fgtr  23806  ufilb  23822  ufilmax  23823  ufilen  23846  flimrest  23899  fclsrest  23940  cfilres  25224  relcmpcmet  25246