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| Mirrors > Home > MPE Home > Th. List > ufilfil | Structured version Visualization version GIF version | ||
| Description: An ultrafilter is a filter. (Contributed by Jeff Hankins, 1-Dec-2009.) (Revised by Mario Carneiro, 29-Jul-2015.) | 
| Ref | Expression | 
|---|---|
| ufilfil | ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | isufil 23912 | . 2 ⊢ (𝐹 ∈ (UFil‘𝑋) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋(𝑥 ∈ 𝐹 ∨ (𝑋 ∖ 𝑥) ∈ 𝐹))) | |
| 2 | 1 | simplbi 497 | 1 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∨ wo 847 ∈ wcel 2107 ∀wral 3060 ∖ cdif 3947 𝒫 cpw 4599 ‘cfv 6560 Filcfil 23854 UFilcufil 23908 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fv 6568 df-ufil 23910 | 
| This theorem is referenced by: ufilb 23915 isufil2 23917 ufprim 23918 trufil 23919 ufileu 23928 filufint 23929 uffixfr 23932 uffix2 23933 uffixsn 23934 uffinfix 23936 cfinufil 23937 ufilen 23939 ufildr 23940 fmufil 23968 ufldom 23971 uffclsflim 24040 ufilcmp 24041 uffcfflf 24048 alexsublem 24053 | 
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