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| Mirrors > Home > MPE Home > Th. List > fbasfip | Structured version Visualization version GIF version | ||
| Description: A filter base has the finite intersection property. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.) |
| Ref | Expression |
|---|---|
| fbasfip | ⊢ (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ (fi‘𝐹)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elin 3967 | . . . . . 6 ⊢ (𝑦 ∈ (𝒫 𝐹 ∩ Fin) ↔ (𝑦 ∈ 𝒫 𝐹 ∧ 𝑦 ∈ Fin)) | |
| 2 | elpwi 4607 | . . . . . . 7 ⊢ (𝑦 ∈ 𝒫 𝐹 → 𝑦 ⊆ 𝐹) | |
| 3 | 2 | anim1i 615 | . . . . . 6 ⊢ ((𝑦 ∈ 𝒫 𝐹 ∧ 𝑦 ∈ Fin) → (𝑦 ⊆ 𝐹 ∧ 𝑦 ∈ Fin)) |
| 4 | 1, 3 | sylbi 217 | . . . . 5 ⊢ (𝑦 ∈ (𝒫 𝐹 ∩ Fin) → (𝑦 ⊆ 𝐹 ∧ 𝑦 ∈ Fin)) |
| 5 | fbssint 23846 | . . . . . 6 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ⊆ 𝐹 ∧ 𝑦 ∈ Fin) → ∃𝑧 ∈ 𝐹 𝑧 ⊆ ∩ 𝑦) | |
| 6 | 5 | 3expb 1121 | . . . . 5 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ (𝑦 ⊆ 𝐹 ∧ 𝑦 ∈ Fin)) → ∃𝑧 ∈ 𝐹 𝑧 ⊆ ∩ 𝑦) |
| 7 | 4, 6 | sylan2 593 | . . . 4 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) → ∃𝑧 ∈ 𝐹 𝑧 ⊆ ∩ 𝑦) |
| 8 | 0nelfb 23839 | . . . . . . . . 9 ⊢ (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐹) | |
| 9 | 8 | ad2antrr 726 | . . . . . . . 8 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ 𝑧 ∈ 𝐹) → ¬ ∅ ∈ 𝐹) |
| 10 | eleq1 2829 | . . . . . . . . . 10 ⊢ (𝑧 = ∅ → (𝑧 ∈ 𝐹 ↔ ∅ ∈ 𝐹)) | |
| 11 | 10 | biimpcd 249 | . . . . . . . . 9 ⊢ (𝑧 ∈ 𝐹 → (𝑧 = ∅ → ∅ ∈ 𝐹)) |
| 12 | 11 | adantl 481 | . . . . . . . 8 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ 𝑧 ∈ 𝐹) → (𝑧 = ∅ → ∅ ∈ 𝐹)) |
| 13 | 9, 12 | mtod 198 | . . . . . . 7 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ 𝑧 ∈ 𝐹) → ¬ 𝑧 = ∅) |
| 14 | ss0 4402 | . . . . . . 7 ⊢ (𝑧 ⊆ ∅ → 𝑧 = ∅) | |
| 15 | 13, 14 | nsyl 140 | . . . . . 6 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ 𝑧 ∈ 𝐹) → ¬ 𝑧 ⊆ ∅) |
| 16 | 15 | adantrr 717 | . . . . 5 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ (𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ ∩ 𝑦)) → ¬ 𝑧 ⊆ ∅) |
| 17 | sseq2 4010 | . . . . . . 7 ⊢ (∅ = ∩ 𝑦 → (𝑧 ⊆ ∅ ↔ 𝑧 ⊆ ∩ 𝑦)) | |
| 18 | 17 | biimprcd 250 | . . . . . 6 ⊢ (𝑧 ⊆ ∩ 𝑦 → (∅ = ∩ 𝑦 → 𝑧 ⊆ ∅)) |
| 19 | 18 | ad2antll 729 | . . . . 5 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ (𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ ∩ 𝑦)) → (∅ = ∩ 𝑦 → 𝑧 ⊆ ∅)) |
| 20 | 16, 19 | mtod 198 | . . . 4 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) ∧ (𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ ∩ 𝑦)) → ¬ ∅ = ∩ 𝑦) |
| 21 | 7, 20 | rexlimddv 3161 | . . 3 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ (𝒫 𝐹 ∩ Fin)) → ¬ ∅ = ∩ 𝑦) |
| 22 | 21 | nrexdv 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)) | |
| 25 | 23, 24 | mpan 690 | . 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → (∅ ∈ (fi‘𝐹) ↔ ∃𝑦 ∈ (𝒫 𝐹 ∩ Fin)∅ = ∩ 𝑦)) |
| 26 | 22, 25 | mtbird 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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