Theorem fbasfip 24192

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 4093	. . . . . 6 ⊢ (𝑦 ∈ (𝒫 𝐹 ∩ Fin) ↔ (𝑦 ∈ 𝒫 𝐹 ∧ 𝑦 ∈ Fin))
2		elpwi 4563	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 𝐹 → 𝑦 ⊆ 𝐹)
3	2	anim1i 627	. . . . . 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 24162	. . . . . 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 605	. . . 4 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) → ∃𝑧 ∈ 𝐹 𝑧 ⊆ ∩ 𝑦)
8		0nelfb 24155	. . . . . . . . 9 ⊢ (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐹)
9	8	ad2antrr 739	. . . . . . . 8 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ 𝑧 ∈ 𝐹) → ¬ ∅ ∈ 𝐹)
10		eleq1 2848	. . . . . . . . . 10 ⊢ (𝑧 = ∅ → (𝑧 ∈ 𝐹 ↔ ∅ ∈ 𝐹))
11	10	biimpcd 252	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐹 → (𝑧 = ∅ → ∅ ∈ 𝐹))
12	11	adantl 487	. . . . . . . 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 4351	. . . . . . 7 ⊢ (𝑧 ⊆ ∅ → 𝑧 = ∅)
15	13, 14	nsyl 141	. . . . . 6 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ 𝑧 ∈ 𝐹) → ¬ 𝑧 ⊆ ∅)
16	15	adantrr 730	. . . . 5 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ (𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ ∩ 𝑦)) → ¬ 𝑧 ⊆ ∅)
17		sseq2 3934	. . . . . . 7 ⊢ (∅ = ∩ 𝑦 → (𝑧 ⊆ ∅ ↔ 𝑧 ⊆ ∩ 𝑦))
18	17	biimprcd 253	. . . . . 6 ⊢ (𝑧 ⊆ ∩ 𝑦 → (∅ = ∩ 𝑦 → 𝑧 ⊆ ∅))
19	18	ad2antll 742	. . . . 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 3169	. . 3 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) → ¬ ∅ = ∩ 𝑦)
22	21	nrexdv 3157	. 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → ¬ ∃𝑦 ∈ (𝒫 𝐹 ∩ Fin)∅ = ∩ 𝑦)
23		0ex 5260	. . 3 ⊢ ∅ ∈ V
24		elfi 9408	. . 3 ⊢ ((∅ ∈ V ∧ 𝐹 ∈ (fBas‘𝑋)) → (∅ ∈ (fi‘𝐹) ↔ ∃𝑦 ∈ (𝒫 𝐹 ∩ Fin)∅ = ∩ 𝑦))
25	23, 24	mpan 703	. 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → (∅ ∈ (fi‘𝐹) ↔ ∃𝑦 ∈ (𝒫 𝐹 ∩ Fin)∅ = ∩ 𝑦))
26	22, 25	mtbird 328	1 ⊢ (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ (fi‘𝐹))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∃wrex 3086  Vcvv 3450   ⊆ wss 3895   ∩ cin 4089  ∅c0 4278  𝒫 cpw 4556  ∩ cint 4906  ‘cfv 6538  Fincfn 8976  ficfi 9405  fBascfbas 21671

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  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 7878  df-1o 8479  df-2o 8480  df-en 8977  df-fin 8980  df-fi 9406  df-fbas 21680

This theorem is used by:  fbunfip  24193