Theorem fbncp 23862

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 23854	. . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐹)
2	1	adantr 480	. 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹) → ¬ ∅ ∈ 𝐹)
3		fbasssin 23859	. . . 4 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ (𝐵 ∖ 𝐴) ∈ 𝐹) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ (𝐵 ∖ 𝐴)))
4		disjdif 4477	. . . . . . . 8 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅
5	4	sseq2i 4024	. . . . . . 7 ⊢ (𝑥 ⊆ (𝐴 ∩ (𝐵 ∖ 𝐴)) ↔ 𝑥 ⊆ ∅)
6		ss0 4407	. . . . . . 7 ⊢ (𝑥 ⊆ ∅ → 𝑥 = ∅)
7	5, 6	sylbi 217	. . . . . 6 ⊢ (𝑥 ⊆ (𝐴 ∩ (𝐵 ∖ 𝐴)) → 𝑥 = ∅)
8		eleq1 2826	. . . . . . 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 3144	. . . 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 1120	. 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 1536   ∈ wcel 2105  ∃wrex 3067   ∖ cdif 3959   ∩ cin 3961   ⊆ wss 3962  ∅c0 4338  ‘cfv 6562  fBascfbas 21369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fv 6570  df-fbas 21378

This theorem is referenced by:  filconn  23906  fgtr  23913  ufilb  23929  ufilmax  23930  ufilen  23953  flimrest  24006  fclsrest  24047  cfilres  25343  relcmpcmet  25365