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Theorem fbasfip 22796
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 3899 . . . . . 6 (𝑦 ∈ (𝒫 𝐹 ∩ Fin) ↔ (𝑦 ∈ 𝒫 𝐹𝑦 ∈ Fin))
2 elpwi 4538 . . . . . . 7 (𝑦 ∈ 𝒫 𝐹𝑦𝐹)
32anim1i 618 . . . . . 6 ((𝑦 ∈ 𝒫 𝐹𝑦 ∈ Fin) → (𝑦𝐹𝑦 ∈ Fin))
41, 3sylbi 220 . . . . 5 (𝑦 ∈ (𝒫 𝐹 ∩ Fin) → (𝑦𝐹𝑦 ∈ Fin))
5 fbssint 22766 . . . . . 6 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦𝐹𝑦 ∈ Fin) → ∃𝑧𝐹 𝑧 𝑦)
653expb 1122 . . . . 5 ((𝐹 ∈ (fBas‘𝑋) ∧ (𝑦𝐹𝑦 ∈ Fin)) → ∃𝑧𝐹 𝑧 𝑦)
74, 6sylan2 596 . . . 4 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) → ∃𝑧𝐹 𝑧 𝑦)
8 0nelfb 22759 . . . . . . . . 9 (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐹)
98ad2antrr 726 . . . . . . . 8 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ 𝑧𝐹) → ¬ ∅ ∈ 𝐹)
10 eleq1 2827 . . . . . . . . . 10 (𝑧 = ∅ → (𝑧𝐹 ↔ ∅ ∈ 𝐹))
1110biimpcd 252 . . . . . . . . 9 (𝑧𝐹 → (𝑧 = ∅ → ∅ ∈ 𝐹))
1211adantl 485 . . . . . . . 8 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ 𝑧𝐹) → (𝑧 = ∅ → ∅ ∈ 𝐹))
139, 12mtod 201 . . . . . . 7 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ 𝑧𝐹) → ¬ 𝑧 = ∅)
14 ss0 4329 . . . . . . 7 (𝑧 ⊆ ∅ → 𝑧 = ∅)
1513, 14nsyl 142 . . . . . 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 3943 . . . . . . 7 (∅ = 𝑦 → (𝑧 ⊆ ∅ ↔ 𝑧 𝑦))
1817biimprcd 253 . . . . . 6 (𝑧 𝑦 → (∅ = 𝑦𝑧 ⊆ ∅))
1918ad2antll 729 . . . . 5 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ (𝑧𝐹𝑧 𝑦)) → (∅ = 𝑦𝑧 ⊆ ∅))
2016, 19mtod 201 . . . 4 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ (𝑧𝐹𝑧 𝑦)) → ¬ ∅ = 𝑦)
217, 20rexlimddv 3220 . . 3 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) → ¬ ∅ = 𝑦)
2221nrexdv 3198 . 2 (𝐹 ∈ (fBas‘𝑋) → ¬ ∃𝑦 ∈ (𝒫 𝐹 ∩ Fin)∅ = 𝑦)
23 0ex 5216 . . 3 ∅ ∈ V
24 elfi 9058 . . 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 328 1 (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ (fi‘𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1543  wcel 2112  wrex 3065  Vcvv 3423  cin 3882  wss 3883  c0 4253  𝒫 cpw 4529   cint 4875  cfv 6400  Fincfn 8649  ficfi 9055  fBascfbas 20383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-sep 5208  ax-nul 5215  ax-pow 5274  ax-pr 5338  ax-un 7544
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3425  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4456  df-pw 4531  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4836  df-int 4876  df-br 5070  df-opab 5132  df-mpt 5152  df-tr 5178  df-id 5471  df-eprel 5477  df-po 5485  df-so 5486  df-fr 5526  df-we 5528  df-xp 5574  df-rel 5575  df-cnv 5576  df-co 5577  df-dm 5578  df-rn 5579  df-res 5580  df-ima 5581  df-ord 6236  df-on 6237  df-lim 6238  df-suc 6239  df-iota 6358  df-fun 6402  df-fn 6403  df-f 6404  df-f1 6405  df-fo 6406  df-f1o 6407  df-fv 6408  df-om 7666  df-1o 8225  df-er 8414  df-en 8650  df-fin 8653  df-fi 9056  df-fbas 20392
This theorem is referenced by:  fbunfip  22797
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