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Theorem fbasfip 23876
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.
StepHypRef Expression
1 elin 3967 . . . . . 6 (𝑦 ∈ (𝒫 𝐹 ∩ Fin) ↔ (𝑦 ∈ 𝒫 𝐹𝑦 ∈ Fin))
2 elpwi 4607 . . . . . . 7 (𝑦 ∈ 𝒫 𝐹𝑦𝐹)
32anim1i 615 . . . . . 6 ((𝑦 ∈ 𝒫 𝐹𝑦 ∈ Fin) → (𝑦𝐹𝑦 ∈ Fin))
41, 3sylbi 217 . . . . 5 (𝑦 ∈ (𝒫 𝐹 ∩ Fin) → (𝑦𝐹𝑦 ∈ Fin))
5 fbssint 23846 . . . . . 6 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦𝐹𝑦 ∈ Fin) → ∃𝑧𝐹 𝑧 𝑦)
653expb 1121 . . . . 5 ((𝐹 ∈ (fBas‘𝑋) ∧ (𝑦𝐹𝑦 ∈ Fin)) → ∃𝑧𝐹 𝑧 𝑦)
74, 6sylan2 593 . . . 4 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) → ∃𝑧𝐹 𝑧 𝑦)
8 0nelfb 23839 . . . . . . . . 9 (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐹)
98ad2antrr 726 . . . . . . . 8 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ 𝑧𝐹) → ¬ ∅ ∈ 𝐹)
10 eleq1 2829 . . . . . . . . . 10 (𝑧 = ∅ → (𝑧𝐹 ↔ ∅ ∈ 𝐹))
1110biimpcd 249 . . . . . . . . 9 (𝑧𝐹 → (𝑧 = ∅ → ∅ ∈ 𝐹))
1211adantl 481 . . . . . . . 8 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ 𝑧𝐹) → (𝑧 = ∅ → ∅ ∈ 𝐹))
139, 12mtod 198 . . . . . . 7 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ 𝑧𝐹) → ¬ 𝑧 = ∅)
14 ss0 4402 . . . . . . 7 (𝑧 ⊆ ∅ → 𝑧 = ∅)
1513, 14nsyl 140 . . . . . 6 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ 𝑧𝐹) → ¬ 𝑧 ⊆ ∅)
1615adantrr 717 . . . . 5 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ (𝑧𝐹𝑧 𝑦)) → ¬ 𝑧 ⊆ ∅)
17 sseq2 4010 . . . . . . 7 (∅ = 𝑦 → (𝑧 ⊆ ∅ ↔ 𝑧 𝑦))
1817biimprcd 250 . . . . . 6 (𝑧 𝑦 → (∅ = 𝑦𝑧 ⊆ ∅))
1918ad2antll 729 . . . . 5 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ (𝑧𝐹𝑧 𝑦)) → (∅ = 𝑦𝑧 ⊆ ∅))
2016, 19mtod 198 . . . 4 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ (𝑧𝐹𝑧 𝑦)) → ¬ ∅ = 𝑦)
217, 20rexlimddv 3161 . . 3 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) → ¬ ∅ = 𝑦)
2221nrexdv 3149 . 2 (𝐹 ∈ (fBas‘𝑋) → ¬ ∃𝑦 ∈ (𝒫 𝐹 ∩ Fin)∅ = 𝑦)
23 0ex 5307 . . 3 ∅ ∈ V
24 elfi 9453 . . 3 ((∅ ∈ V ∧ 𝐹 ∈ (fBas‘𝑋)) → (∅ ∈ (fi‘𝐹) ↔ ∃𝑦 ∈ (𝒫 𝐹 ∩ Fin)∅ = 𝑦))
2523, 24mpan 690 . 2 (𝐹 ∈ (fBas‘𝑋) → (∅ ∈ (fi‘𝐹) ↔ ∃𝑦 ∈ (𝒫 𝐹 ∩ Fin)∅ = 𝑦))
2622, 25mtbird 325 1 (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ (fi‘𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wrex 3070  Vcvv 3480  cin 3950  wss 3951  c0 4333  𝒫 cpw 4600   cint 4946  cfv 6561  Fincfn 8985  ficfi 9450  fBascfbas 21352
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-1o 8506  df-2o 8507  df-en 8986  df-fin 8989  df-fi 9451  df-fbas 21361
This theorem is referenced by:  fbunfip  23877
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