Theorem fbasfip 24006

Description: A filter base has the finite intersection property. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Assertion

Ref	Expression
fbasfip	⊢ (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ (fi‘𝐹))

Proof of Theorem fbasfip

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3922	. . . . . 6 ⊢ (𝑦 ∈ (𝒫 𝐹 ∩ Fin) ↔ (𝑦 ∈ 𝒫 𝐹 ∧ 𝑦 ∈ Fin))
2		elpwi 4570	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 𝐹 → 𝑦 ⊆ 𝐹)
3	2	anim1i 626	. . . . . 6 ⊢ ((𝑦 ∈ 𝒫 𝐹 ∧ 𝑦 ∈ Fin) → (𝑦 ⊆ 𝐹 ∧ 𝑦 ∈ Fin))
4	1, 3	sylbi 220	. . . . 5 ⊢ (𝑦 ∈ (𝒫 𝐹 ∩ Fin) → (𝑦 ⊆ 𝐹 ∧ 𝑦 ∈ Fin))
5		fbssint 23976	. . . . . 6 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ⊆ 𝐹 ∧ 𝑦 ∈ Fin) → ∃𝑧 ∈ 𝐹 𝑧 ⊆ ∩ 𝑦)
6	5	3expb 1138	. . . . 5 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ (𝑦 ⊆ 𝐹 ∧ 𝑦 ∈ Fin)) → ∃𝑧 ∈ 𝐹 𝑧 ⊆ ∩ 𝑦)
7	4, 6	sylan2 604	. . . 4 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) → ∃𝑧 ∈ 𝐹 𝑧 ⊆ ∩ 𝑦)
8		0nelfb 23969	. . . . . . . . 9 ⊢ (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐹)
9	8	ad2antrr 738	. . . . . . . 8 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ 𝑧 ∈ 𝐹) → ¬ ∅ ∈ 𝐹)
10		eleq1 2851	. . . . . . . . . 10 ⊢ (𝑧 = ∅ → (𝑧 ∈ 𝐹 ↔ ∅ ∈ 𝐹))
11	10	biimpcd 252	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐹 → (𝑧 = ∅ → ∅ ∈ 𝐹))
12	11	adantl 486	. . . . . . . 8 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ 𝑧 ∈ 𝐹) → (𝑧 = ∅ → ∅ ∈ 𝐹))
13	9, 12	mtod 201	. . . . . . 7 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ 𝑧 ∈ 𝐹) → ¬ 𝑧 = ∅)
14		ss0 4360	. . . . . . 7 ⊢ (𝑧 ⊆ ∅ → 𝑧 = ∅)
15	13, 14	nsyl 141	. . . . . 6 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ 𝑧 ∈ 𝐹) → ¬ 𝑧 ⊆ ∅)
16	15	adantrr 729	. . . . 5 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ (𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ ∩ 𝑦)) → ¬ 𝑧 ⊆ ∅)
17		sseq2 3964	. . . . . . 7 ⊢ (∅ = ∩ 𝑦 → (𝑧 ⊆ ∅ ↔ 𝑧 ⊆ ∩ 𝑦))
18	17	biimprcd 253	. . . . . 6 ⊢ (𝑧 ⊆ ∩ 𝑦 → (∅ = ∩ 𝑦 → 𝑧 ⊆ ∅))
19	18	ad2antll 741	. . . . 5 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ (𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ ∩ 𝑦)) → (∅ = ∩ 𝑦 → 𝑧 ⊆ ∅))
20	16, 19	mtod 201	. . . 4 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ (𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ ∩ 𝑦)) → ¬ ∅ = ∩ 𝑦)
21	7, 20	rexlimddv 3172	. . 3 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) → ¬ ∅ = ∩ 𝑦)
22	21	nrexdv 3160	. 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → ¬ ∃𝑦 ∈ (𝒫 𝐹 ∩ Fin)∅ = ∩ 𝑦)
23		0ex 5271	. . 3 ⊢ ∅ ∈ V
24		elfi 9374	. . 3 ⊢ ((∅ ∈ V ∧ 𝐹 ∈ (fBas‘𝑋)) → (∅ ∈ (fi‘𝐹) ↔ ∃𝑦 ∈ (𝒫 𝐹 ∩ Fin)∅ = ∩ 𝑦))
25	23, 24	mpan 702	. 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → (∅ ∈ (fi‘𝐹) ↔ ∃𝑦 ∈ (𝒫 𝐹 ∩ Fin)∅ = ∩ 𝑦))
26	22, 25	mtbird 328	1 ⊢ (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ (fi‘𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∃wrex 3089  Vcvv 3455   ∩ cin 3905   ⊆ wss 3906  ∅c0 4287  𝒫 cpw 4563  ∩ cint 4913  ‘cfv 6538  Fincfn 8944  ficfi 9371  fBascfbas 21491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-om 7864  df-1o 8454  df-2o 8455  df-en 8945  df-fin 8948  df-fi 9372  df-fbas 21500

This theorem is referenced by:  fbunfip  24007