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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.
StepHypRef Expression
1 elin 3956 . . . . . 6 (𝑦 ∈ (𝒫 𝐹 ∩ Fin) ↔ (𝑦 ∈ 𝒫 𝐹𝑦 ∈ Fin))
2 elpwi 4601 . . . . . . 7 (𝑦 ∈ 𝒫 𝐹𝑦𝐹)
32anim1i 614 . . . . . 6 ((𝑦 ∈ 𝒫 𝐹𝑦 ∈ Fin) → (𝑦𝐹𝑦 ∈ Fin))
41, 3sylbi 216 . . . . 5 (𝑦 ∈ (𝒫 𝐹 ∩ Fin) → (𝑦𝐹𝑦 ∈ Fin))
5 fbssint 23663 . . . . . 6 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦𝐹𝑦 ∈ Fin) → ∃𝑧𝐹 𝑧 𝑦)
653expb 1117 . . . . 5 ((𝐹 ∈ (fBas‘𝑋) ∧ (𝑦𝐹𝑦 ∈ Fin)) → ∃𝑧𝐹 𝑧 𝑦)
74, 6sylan2 592 . . . 4 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) → ∃𝑧𝐹 𝑧 𝑦)
8 0nelfb 23656 . . . . . . . . 9 (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐹)
98ad2antrr 723 . . . . . . . 8 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ 𝑧𝐹) → ¬ ∅ ∈ 𝐹)
10 eleq1 2813 . . . . . . . . . 10 (𝑧 = ∅ → (𝑧𝐹 ↔ ∅ ∈ 𝐹))
1110biimpcd 248 . . . . . . . . 9 (𝑧𝐹 → (𝑧 = ∅ → ∅ ∈ 𝐹))
1211adantl 481 . . . . . . . 8 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ 𝑧𝐹) → (𝑧 = ∅ → ∅ ∈ 𝐹))
139, 12mtod 197 . . . . . . 7 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ 𝑧𝐹) → ¬ 𝑧 = ∅)
14 ss0 4390 . . . . . . 7 (𝑧 ⊆ ∅ → 𝑧 = ∅)
1513, 14nsyl 140 . . . . . 6 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ 𝑧𝐹) → ¬ 𝑧 ⊆ ∅)
1615adantrr 714 . . . . 5 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ (𝑧𝐹𝑧 𝑦)) → ¬ 𝑧 ⊆ ∅)
17 sseq2 4000 . . . . . . 7 (∅ = 𝑦 → (𝑧 ⊆ ∅ ↔ 𝑧 𝑦))
1817biimprcd 249 . . . . . 6 (𝑧 𝑦 → (∅ = 𝑦𝑧 ⊆ ∅))
1918ad2antll 726 . . . . 5 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ (𝑧𝐹𝑧 𝑦)) → (∅ = 𝑦𝑧 ⊆ ∅))
2016, 19mtod 197 . . . 4 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ (𝑧𝐹𝑧 𝑦)) → ¬ ∅ = 𝑦)
217, 20rexlimddv 3153 . . 3 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) → ¬ ∅ = 𝑦)
2221nrexdv 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))
2523, 24mpan 687 . 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 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
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