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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 22513 | . 2 ⊢ (𝐹 ∈ (UFil‘𝑋) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋(𝑥 ∈ 𝐹 ∨ (𝑋 ∖ 𝑥) ∈ 𝐹))) | |
2 | 1 | simplbi 500 | 1 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 843 ∈ wcel 2114 ∀wral 3140 ∖ cdif 3935 𝒫 cpw 4541 ‘cfv 6357 Filcfil 22455 UFilcufil 22509 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fv 6365 df-ufil 22511 |
This theorem is referenced by: ufilb 22516 isufil2 22518 ufprim 22519 trufil 22520 ufileu 22529 filufint 22530 uffixfr 22533 uffix2 22534 uffixsn 22535 uffinfix 22537 cfinufil 22538 ufilen 22540 ufildr 22541 fmufil 22569 ufldom 22572 uffclsflim 22641 ufilcmp 22642 uffcfflf 22649 alexsublem 22654 |
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