Theorem fbasfip 23771

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 3921	. . . . . 6 ⊢ (𝑦 ∈ (𝒫 𝐹 ∩ Fin) ↔ (𝑦 ∈ 𝒫 𝐹 ∧ 𝑦 ∈ Fin))
2		elpwi 4560	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 𝐹 → 𝑦 ⊆ 𝐹)
3	2	anim1i 615	. . . . . 6 ⊢ ((𝑦 ∈ 𝒫 𝐹 ∧ 𝑦 ∈ Fin) → (𝑦 ⊆ 𝐹 ∧ 𝑦 ∈ Fin))
4	1, 3	sylbi 217	. . . . 5 ⊢ (𝑦 ∈ (𝒫 𝐹 ∩ Fin) → (𝑦 ⊆ 𝐹 ∧ 𝑦 ∈ Fin))
5		fbssint 23741	. . . . . 6 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ⊆ 𝐹 ∧ 𝑦 ∈ Fin) → ∃𝑧 ∈ 𝐹 𝑧 ⊆ ∩ 𝑦)
6	5	3expb 1120	. . . . 5 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ (𝑦 ⊆ 𝐹 ∧ 𝑦 ∈ Fin)) → ∃𝑧 ∈ 𝐹 𝑧 ⊆ ∩ 𝑦)
7	4, 6	sylan2 593	. . . 4 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) → ∃𝑧 ∈ 𝐹 𝑧 ⊆ ∩ 𝑦)
8		0nelfb 23734	. . . . . . . . 9 ⊢ (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐹)
9	8	ad2antrr 726	. . . . . . . 8 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ 𝑧 ∈ 𝐹) → ¬ ∅ ∈ 𝐹)
10		eleq1 2816	. . . . . . . . . 10 ⊢ (𝑧 = ∅ → (𝑧 ∈ 𝐹 ↔ ∅ ∈ 𝐹))
11	10	biimpcd 249	. . . . . . . . 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 198	. . . . . . 7 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ 𝑧 ∈ 𝐹) → ¬ 𝑧 = ∅)
14		ss0 4355	. . . . . . 7 ⊢ (𝑧 ⊆ ∅ → 𝑧 = ∅)
15	13, 14	nsyl 140	. . . . . 6 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ 𝑧 ∈ 𝐹) → ¬ 𝑧 ⊆ ∅)
16	15	adantrr 717	. . . . 5 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ (𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ ∩ 𝑦)) → ¬ 𝑧 ⊆ ∅)
17		sseq2 3964	. . . . . . 7 ⊢ (∅ = ∩ 𝑦 → (𝑧 ⊆ ∅ ↔ 𝑧 ⊆ ∩ 𝑦))
18	17	biimprcd 250	. . . . . 6 ⊢ (𝑧 ⊆ ∩ 𝑦 → (∅ = ∩ 𝑦 → 𝑧 ⊆ ∅))
19	18	ad2antll 729	. . . . 5 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ (𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ ∩ 𝑦)) → (∅ = ∩ 𝑦 → 𝑧 ⊆ ∅))
20	16, 19	mtod 198	. . . 4 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ (𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ ∩ 𝑦)) → ¬ ∅ = ∩ 𝑦)
21	7, 20	rexlimddv 3136	. . 3 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) → ¬ ∅ = ∩ 𝑦)
22	21	nrexdv 3124	. 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → ¬ ∃𝑦 ∈ (𝒫 𝐹 ∩ Fin)∅ = ∩ 𝑦)
23		0ex 5249	. . 3 ⊢ ∅ ∈ V
24		elfi 9322	. . 3 ⊢ ((∅ ∈ V ∧ 𝐹 ∈ (fBas‘𝑋)) → (∅ ∈ (fi‘𝐹) ↔ ∃𝑦 ∈ (𝒫 𝐹 ∩ Fin)∅ = ∩ 𝑦))
25	23, 24	mpan 690	. 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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  Vcvv 3438   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286  𝒫 cpw 4553  ∩ cint 4899  ‘cfv 6486  Fincfn 8879  ficfi 9319  fBascfbas 21267

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-om 7807  df-1o 8395  df-2o 8396  df-en 8880  df-fin 8883  df-fi 9320  df-fbas 21276

This theorem is referenced by:  fbunfip  23772