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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 23872 | . . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋)) | |
2 | fbasne0 23854 | . . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ≠ ∅) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅) |
4 | n0 4359 | . . 3 ⊢ (𝐹 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐹) | |
5 | filelss 23876 | . . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐹) → 𝑥 ⊆ 𝑋) | |
6 | ssid 4018 | . . . . . . 7 ⊢ 𝑋 ⊆ 𝑋 | |
7 | filss 23877 | . . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ∈ 𝐹 ∧ 𝑋 ⊆ 𝑋 ∧ 𝑥 ⊆ 𝑋)) → 𝑋 ∈ 𝐹) | |
8 | 7 | 3exp2 1353 | . . . . . . . 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 1931 | . . 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 1776 ∈ wcel 2106 ≠ wne 2938 ⊆ wss 3963 ∅c0 4339 ‘cfv 6563 fBascfbas 21370 Filcfil 23869 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fv 6571 df-fbas 21379 df-fil 23870 |
This theorem is referenced by: isfil2 23880 filn0 23886 infil 23887 filunibas 23905 filuni 23909 trfil1 23910 trfil2 23911 fgtr 23914 trfg 23915 isufil2 23932 filssufil 23936 ssufl 23942 ufileu 23943 filufint 23944 uffixfr 23947 cfinufil 23952 rnelfmlem 23976 rnelfm 23977 fmfnfmlem1 23978 fmfnfmlem2 23979 fmfnfmlem4 23981 fmfnfm 23982 flfval 24014 fclsfnflim 24051 flimfnfcls 24052 fcfval 24057 alexsublem 24068 metust 24587 cmetss 25364 minveclem4a 25478 filnetlem3 36363 filnetlem4 36364 |
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