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Theorem fbasfip 23792
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 3965 . . . . . 6 (𝑦 ∈ (𝒫 𝐹 ∩ Fin) ↔ (𝑦 ∈ 𝒫 𝐹𝑦 ∈ Fin))
2 elpwi 4613 . . . . . . 7 (𝑦 ∈ 𝒫 𝐹𝑦𝐹)
32anim1i 613 . . . . . 6 ((𝑦 ∈ 𝒫 𝐹𝑦 ∈ Fin) → (𝑦𝐹𝑦 ∈ Fin))
41, 3sylbi 216 . . . . 5 (𝑦 ∈ (𝒫 𝐹 ∩ Fin) → (𝑦𝐹𝑦 ∈ Fin))
5 fbssint 23762 . . . . . 6 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦𝐹𝑦 ∈ Fin) → ∃𝑧𝐹 𝑧 𝑦)
653expb 1117 . . . . 5 ((𝐹 ∈ (fBas‘𝑋) ∧ (𝑦𝐹𝑦 ∈ Fin)) → ∃𝑧𝐹 𝑧 𝑦)
74, 6sylan2 591 . . . 4 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) → ∃𝑧𝐹 𝑧 𝑦)
8 0nelfb 23755 . . . . . . . . 9 (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐹)
98ad2antrr 724 . . . . . . . 8 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ 𝑧𝐹) → ¬ ∅ ∈ 𝐹)
10 eleq1 2817 . . . . . . . . . 10 (𝑧 = ∅ → (𝑧𝐹 ↔ ∅ ∈ 𝐹))
1110biimpcd 248 . . . . . . . . 9 (𝑧𝐹 → (𝑧 = ∅ → ∅ ∈ 𝐹))
1211adantl 480 . . . . . . . 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 4402 . . . . . . 7 (𝑧 ⊆ ∅ → 𝑧 = ∅)
1513, 14nsyl 140 . . . . . 6 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ 𝑧𝐹) → ¬ 𝑧 ⊆ ∅)
1615adantrr 715 . . . . 5 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ (𝑧𝐹𝑧 𝑦)) → ¬ 𝑧 ⊆ ∅)
17 sseq2 4008 . . . . . . 7 (∅ = 𝑦 → (𝑧 ⊆ ∅ ↔ 𝑧 𝑦))
1817biimprcd 249 . . . . . 6 (𝑧 𝑦 → (∅ = 𝑦𝑧 ⊆ ∅))
1918ad2antll 727 . . . . 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 3158 . . 3 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) → ¬ ∅ = 𝑦)
2221nrexdv 3146 . 2 (𝐹 ∈ (fBas‘𝑋) → ¬ ∃𝑦 ∈ (𝒫 𝐹 ∩ Fin)∅ = 𝑦)
23 0ex 5311 . . 3 ∅ ∈ V
24 elfi 9444 . . 3 ((∅ ∈ V ∧ 𝐹 ∈ (fBas‘𝑋)) → (∅ ∈ (fi‘𝐹) ↔ ∃𝑦 ∈ (𝒫 𝐹 ∩ Fin)∅ = 𝑦))
2523, 24mpan 688 . 2 (𝐹 ∈ (fBas‘𝑋) → (∅ ∈ (fi‘𝐹) ↔ ∃𝑦 ∈ (𝒫 𝐹 ∩ Fin)∅ = 𝑦))
2622, 25mtbird 324 1 (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ (fi‘𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wrex 3067  Vcvv 3473  cin 3948  wss 3949  c0 4326  𝒫 cpw 4606   cint 4953  cfv 6553  Fincfn 8970  ficfi 9441  fBascfbas 21274
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-om 7877  df-1o 8493  df-er 8731  df-en 8971  df-fin 8974  df-fi 9442  df-fbas 21283
This theorem is referenced by:  fbunfip  23793
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