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Theorem fbasfip 21582
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 3774 . . . . . 6 (𝑦 ∈ (𝒫 𝐹 ∩ Fin) ↔ (𝑦 ∈ 𝒫 𝐹𝑦 ∈ Fin))
2 elpwi 4140 . . . . . . 7 (𝑦 ∈ 𝒫 𝐹𝑦𝐹)
32anim1i 591 . . . . . 6 ((𝑦 ∈ 𝒫 𝐹𝑦 ∈ Fin) → (𝑦𝐹𝑦 ∈ Fin))
41, 3sylbi 207 . . . . 5 (𝑦 ∈ (𝒫 𝐹 ∩ Fin) → (𝑦𝐹𝑦 ∈ Fin))
5 fbssint 21552 . . . . . 6 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦𝐹𝑦 ∈ Fin) → ∃𝑧𝐹 𝑧 𝑦)
653expb 1263 . . . . 5 ((𝐹 ∈ (fBas‘𝑋) ∧ (𝑦𝐹𝑦 ∈ Fin)) → ∃𝑧𝐹 𝑧 𝑦)
74, 6sylan2 491 . . . 4 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) → ∃𝑧𝐹 𝑧 𝑦)
8 0nelfb 21545 . . . . . . . . 9 (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐹)
98ad2antrr 761 . . . . . . . 8 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ 𝑧𝐹) → ¬ ∅ ∈ 𝐹)
10 eleq1 2686 . . . . . . . . . 10 (𝑧 = ∅ → (𝑧𝐹 ↔ ∅ ∈ 𝐹))
1110biimpcd 239 . . . . . . . . 9 (𝑧𝐹 → (𝑧 = ∅ → ∅ ∈ 𝐹))
1211adantl 482 . . . . . . . 8 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ 𝑧𝐹) → (𝑧 = ∅ → ∅ ∈ 𝐹))
139, 12mtod 189 . . . . . . 7 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ 𝑧𝐹) → ¬ 𝑧 = ∅)
14 ss0 3946 . . . . . . 7 (𝑧 ⊆ ∅ → 𝑧 = ∅)
1513, 14nsyl 135 . . . . . 6 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ 𝑧𝐹) → ¬ 𝑧 ⊆ ∅)
1615adantrr 752 . . . . 5 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ (𝑧𝐹𝑧 𝑦)) → ¬ 𝑧 ⊆ ∅)
17 sseq2 3606 . . . . . . 7 (∅ = 𝑦 → (𝑧 ⊆ ∅ ↔ 𝑧 𝑦))
1817biimprcd 240 . . . . . 6 (𝑧 𝑦 → (∅ = 𝑦𝑧 ⊆ ∅))
1918ad2antll 764 . . . . 5 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ (𝑧𝐹𝑧 𝑦)) → (∅ = 𝑦𝑧 ⊆ ∅))
2016, 19mtod 189 . . . 4 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ (𝑧𝐹𝑧 𝑦)) → ¬ ∅ = 𝑦)
217, 20rexlimddv 3028 . . 3 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) → ¬ ∅ = 𝑦)
2221nrexdv 2995 . 2 (𝐹 ∈ (fBas‘𝑋) → ¬ ∃𝑦 ∈ (𝒫 𝐹 ∩ Fin)∅ = 𝑦)
23 0ex 4750 . . 3 ∅ ∈ V
24 elfi 8263 . . 3 ((∅ ∈ V ∧ 𝐹 ∈ (fBas‘𝑋)) → (∅ ∈ (fi‘𝐹) ↔ ∃𝑦 ∈ (𝒫 𝐹 ∩ Fin)∅ = 𝑦))
2523, 24mpan 705 . 2 (𝐹 ∈ (fBas‘𝑋) → (∅ ∈ (fi‘𝐹) ↔ ∃𝑦 ∈ (𝒫 𝐹 ∩ Fin)∅ = 𝑦))
2622, 25mtbird 315 1 (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ (fi‘𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wrex 2908  Vcvv 3186  cin 3554  wss 3555  c0 3891  𝒫 cpw 4130   cint 4440  cfv 5847  Fincfn 7899  ficfi 8260  fBascfbas 19653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-en 7900  df-fin 7903  df-fi 8261  df-fbas 19662
This theorem is referenced by:  fbunfip  21583
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