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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 22489 | . 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 3133 ∖ cdif 3916 𝒫 cpw 4520 ‘cfv 6336 Filcfil 22431 UFilcufil 22485 |
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 2792 ax-sep 5184 ax-nul 5191 ax-pow 5247 ax-pr 5311 |
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 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ral 3138 df-rex 3139 df-rab 3142 df-v 3483 df-sbc 3759 df-csb 3867 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-nul 4275 df-if 4449 df-pw 4522 df-sn 4549 df-pr 4551 df-op 4555 df-uni 4820 df-br 5048 df-opab 5110 df-mpt 5128 df-id 5441 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6295 df-fun 6338 df-fv 6344 df-ufil 22487 |
This theorem is referenced by: ufilb 22492 isufil2 22494 ufprim 22495 trufil 22496 ufileu 22505 filufint 22506 uffixfr 22509 uffix2 22510 uffixsn 22511 uffinfix 22513 cfinufil 22514 ufilen 22516 ufildr 22517 fmufil 22545 ufldom 22548 uffclsflim 22617 ufilcmp 22618 uffcfflf 22625 alexsublem 22630 |
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