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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 22060 | . . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋)) | |
2 | fbasne0 22042 | . . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ≠ ∅) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅) |
4 | n0 4159 | . . 3 ⊢ (𝐹 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐹) | |
5 | filelss 22064 | . . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐹) → 𝑥 ⊆ 𝑋) | |
6 | ssid 3842 | . . . . . . 7 ⊢ 𝑋 ⊆ 𝑋 | |
7 | filss 22065 | . . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ∈ 𝐹 ∧ 𝑋 ⊆ 𝑋 ∧ 𝑥 ⊆ 𝑋)) → 𝑋 ∈ 𝐹) | |
8 | 7 | 3exp2 1416 | . . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑥 ∈ 𝐹 → (𝑋 ⊆ 𝑋 → (𝑥 ⊆ 𝑋 → 𝑋 ∈ 𝐹)))) |
9 | 8 | imp 397 | . . . . . . 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 403 | . . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑥 ∈ 𝐹 → 𝑋 ∈ 𝐹)) |
13 | 12 | exlimdv 1976 | . . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → (∃𝑥 𝑥 ∈ 𝐹 → 𝑋 ∈ 𝐹)) |
14 | 4, 13 | syl5bi 234 | . 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 386 ∃wex 1823 ∈ wcel 2107 ≠ wne 2969 ⊆ wss 3792 ∅c0 4141 ‘cfv 6135 fBascfbas 20130 Filcfil 22057 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fv 6143 df-fbas 20139 df-fil 22058 |
This theorem is referenced by: isfil2 22068 filn0 22074 infil 22075 filunibas 22093 filuni 22097 trfil1 22098 trfil2 22099 fgtr 22102 trfg 22103 isufil2 22120 filssufil 22124 ssufl 22130 ufileu 22131 filufint 22132 uffixfr 22135 cfinufil 22140 rnelfmlem 22164 rnelfm 22165 fmfnfmlem1 22166 fmfnfmlem2 22167 fmfnfmlem4 22169 fmfnfm 22170 flfval 22202 fclsfnflim 22239 flimfnfcls 22240 fcfval 22245 alexsublem 22256 metust 22771 cmetss 23522 minveclem4a 23636 filnetlem3 32963 filnetlem4 32964 |
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