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| Mirrors > Home > MPE Home > Th. List > filtop | Structured version Visualization version GIF version | ||
| Description: The underlying set belongs to the filter. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.) |
| Ref | Expression |
|---|---|
| filtop | ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | filfbas 23856 | . . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋)) | |
| 2 | fbasne0 23838 | . . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ≠ ∅) | |
| 3 | 1, 2 | syl 17 | . 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅) |
| 4 | n0 4353 | . . 3 ⊢ (𝐹 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐹) | |
| 5 | filelss 23860 | . . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐹) → 𝑥 ⊆ 𝑋) | |
| 6 | ssid 4006 | . . . . . . 7 ⊢ 𝑋 ⊆ 𝑋 | |
| 7 | filss 23861 | . . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ∈ 𝐹 ∧ 𝑋 ⊆ 𝑋 ∧ 𝑥 ⊆ 𝑋)) → 𝑋 ∈ 𝐹) | |
| 8 | 7 | 3exp2 1355 | . . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑥 ∈ 𝐹 → (𝑋 ⊆ 𝑋 → (𝑥 ⊆ 𝑋 → 𝑋 ∈ 𝐹)))) |
| 9 | 8 | imp 406 | . . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐹) → (𝑋 ⊆ 𝑋 → (𝑥 ⊆ 𝑋 → 𝑋 ∈ 𝐹))) |
| 10 | 6, 9 | mpi 20 | . . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐹) → (𝑥 ⊆ 𝑋 → 𝑋 ∈ 𝐹)) |
| 11 | 5, 10 | mpd 15 | . . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐹) → 𝑋 ∈ 𝐹) |
| 12 | 11 | ex 412 | . . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑥 ∈ 𝐹 → 𝑋 ∈ 𝐹)) |
| 13 | 12 | exlimdv 1933 | . . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → (∃𝑥 𝑥 ∈ 𝐹 → 𝑋 ∈ 𝐹)) |
| 14 | 4, 13 | biimtrid 242 | . 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 ≠ ∅ → 𝑋 ∈ 𝐹)) |
| 15 | 3, 14 | mpd 15 | 1 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∃wex 1779 ∈ wcel 2108 ≠ wne 2940 ⊆ wss 3951 ∅c0 4333 ‘cfv 6561 fBascfbas 21352 Filcfil 23853 |
| 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 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 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-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 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-iota 6514 df-fun 6563 df-fv 6569 df-fbas 21361 df-fil 23854 |
| This theorem is referenced by: isfil2 23864 filn0 23870 infil 23871 filunibas 23889 filuni 23893 trfil1 23894 trfil2 23895 fgtr 23898 trfg 23899 isufil2 23916 filssufil 23920 ssufl 23926 ufileu 23927 filufint 23928 uffixfr 23931 cfinufil 23936 rnelfmlem 23960 rnelfm 23961 fmfnfmlem1 23962 fmfnfmlem2 23963 fmfnfmlem4 23965 fmfnfm 23966 flfval 23998 fclsfnflim 24035 flimfnfcls 24036 fcfval 24041 alexsublem 24052 metust 24571 cmetss 25350 minveclem4a 25464 filnetlem3 36381 filnetlem4 36382 |
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