Theorem fbasfip 23693

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 3956	. . . . . 6 ⊢ (𝑦 ∈ (𝒫 𝐹 ∩ Fin) ↔ (𝑦 ∈ 𝒫 𝐹 ∧ 𝑦 ∈ Fin))
2		elpwi 4601	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 𝐹 → 𝑦 ⊆ 𝐹)
3	2	anim1i 614	. . . . . 6 ⊢ ((𝑦 ∈ 𝒫 𝐹 ∧ 𝑦 ∈ Fin) → (𝑦 ⊆ 𝐹 ∧ 𝑦 ∈ Fin))
4	1, 3	sylbi 216	. . . . 5 ⊢ (𝑦 ∈ (𝒫 𝐹 ∩ Fin) → (𝑦 ⊆ 𝐹 ∧ 𝑦 ∈ Fin))
5		fbssint 23663	. . . . . 6 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ⊆ 𝐹 ∧ 𝑦 ∈ Fin) → ∃𝑧 ∈ 𝐹 𝑧 ⊆ ∩ 𝑦)
6	5	3expb 1117	. . . . 5 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ (𝑦 ⊆ 𝐹 ∧ 𝑦 ∈ Fin)) → ∃𝑧 ∈ 𝐹 𝑧 ⊆ ∩ 𝑦)
7	4, 6	sylan2 592	. . . 4 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) → ∃𝑧 ∈ 𝐹 𝑧 ⊆ ∩ 𝑦)
8		0nelfb 23656	. . . . . . . . 9 ⊢ (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐹)
9	8	ad2antrr 723	. . . . . . . 8 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ 𝑧 ∈ 𝐹) → ¬ ∅ ∈ 𝐹)
10		eleq1 2813	. . . . . . . . . 10 ⊢ (𝑧 = ∅ → (𝑧 ∈ 𝐹 ↔ ∅ ∈ 𝐹))
11	10	biimpcd 248	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐹 → (𝑧 = ∅ → ∅ ∈ 𝐹))
12	11	adantl 481	. . . . . . . 8 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ 𝑧 ∈ 𝐹) → (𝑧 = ∅ → ∅ ∈ 𝐹))
13	9, 12	mtod 197	. . . . . . 7 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ 𝑧 ∈ 𝐹) → ¬ 𝑧 = ∅)
14		ss0 4390	. . . . . . 7 ⊢ (𝑧 ⊆ ∅ → 𝑧 = ∅)
15	13, 14	nsyl 140	. . . . . 6 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ 𝑧 ∈ 𝐹) → ¬ 𝑧 ⊆ ∅)
16	15	adantrr 714	. . . . 5 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ (𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ ∩ 𝑦)) → ¬ 𝑧 ⊆ ∅)
17		sseq2 4000	. . . . . . 7 ⊢ (∅ = ∩ 𝑦 → (𝑧 ⊆ ∅ ↔ 𝑧 ⊆ ∩ 𝑦))
18	17	biimprcd 249	. . . . . 6 ⊢ (𝑧 ⊆ ∩ 𝑦 → (∅ = ∩ 𝑦 → 𝑧 ⊆ ∅))
19	18	ad2antll 726	. . . . 5 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ (𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ ∩ 𝑦)) → (∅ = ∩ 𝑦 → 𝑧 ⊆ ∅))
20	16, 19	mtod 197	. . . 4 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ (𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ ∩ 𝑦)) → ¬ ∅ = ∩ 𝑦)
21	7, 20	rexlimddv 3153	. . 3 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) → ¬ ∅ = ∩ 𝑦)
22	21	nrexdv 3141	. 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → ¬ ∃𝑦 ∈ (𝒫 𝐹 ∩ Fin)∅ = ∩ 𝑦)
23		0ex 5297	. . 3 ⊢ ∅ ∈ V
24		elfi 9403	. . 3 ⊢ ((∅ ∈ V ∧ 𝐹 ∈ (fBas‘𝑋)) → (∅ ∈ (fi‘𝐹) ↔ ∃𝑦 ∈ (𝒫 𝐹 ∩ Fin)∅ = ∩ 𝑦))
25	23, 24	mpan 687	. 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → (∅ ∈ (fi‘𝐹) ↔ ∃𝑦 ∈ (𝒫 𝐹 ∩ Fin)∅ = ∩ 𝑦))
26	22, 25	mtbird 325	1 ⊢ (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ (fi‘𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ∃wrex 3062  Vcvv 3466   ∩ cin 3939   ⊆ wss 3940  ∅c0 4314  𝒫 cpw 4594  ∩ cint 4940  ‘cfv 6533  Fincfn 8934  ficfi 9400  fBascfbas 21215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-om 7849  df-1o 8461  df-er 8698  df-en 8935  df-fin 8938  df-fi 9401  df-fbas 21224

This theorem is referenced by:  fbunfip  23694