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| Mirrors > Home > MPE Home > Th. List > filelss | Structured version Visualization version GIF version | ||
| Description: An element of a filter is a subset of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.) | 
| Ref | Expression | 
|---|---|
| filelss | ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ 𝑋) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | filfbas 23856 | . 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋)) | |
| 2 | fbelss 23841 | . 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ 𝑋) | |
| 3 | 1, 2 | sylan 580 | 1 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ 𝑋) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ⊆ wss 3951 ‘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: filin 23862 filtop 23863 filuni 23893 trfil2 23895 trfil3 23896 fgtr 23898 trfg 23899 ufilmax 23915 isufil2 23916 ufileu 23927 filufint 23928 cfinufil 23936 ufilen 23938 rnelfm 23961 fmfnfmlem4 23965 fmid 23968 flimclsi 23986 flimrest 23991 txflf 24014 fclsopn 24022 fclsrest 24032 flimfnfcls 24036 fclscmpi 24037 iscfil2 25300 cfil3i 25303 iscmet3lem2 25326 iscmet3 25327 cfilresi 25329 cfilres 25330 filnetlem3 36381 | 
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