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Mirrors > Home > MPE Home > Th. List > filssufil | Structured version Visualization version GIF version |
Description: A filter is contained in some ultrafilter. (Requires the Axiom of Choice, via numth3 9688.) (Contributed by Jeff Hankins, 2-Dec-2009.) (Revised by Stefan O'Rear, 29-Jul-2015.) |
Ref | Expression |
---|---|
filssufil | ⊢ (𝐹 ∈ (Fil‘𝑋) → ∃𝑓 ∈ (UFil‘𝑋)𝐹 ⊆ 𝑓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | filtop 22182 | . . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹) | |
2 | pwexg 5128 | . . 3 ⊢ (𝑋 ∈ 𝐹 → 𝒫 𝑋 ∈ V) | |
3 | pwexg 5128 | . . 3 ⊢ (𝒫 𝑋 ∈ V → 𝒫 𝒫 𝑋 ∈ V) | |
4 | numth3 9688 | . . 3 ⊢ (𝒫 𝒫 𝑋 ∈ V → 𝒫 𝒫 𝑋 ∈ dom card) | |
5 | 1, 2, 3, 4 | 4syl 19 | . 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝒫 𝒫 𝑋 ∈ dom card) |
6 | filssufilg 22238 | . 2 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → ∃𝑓 ∈ (UFil‘𝑋)𝐹 ⊆ 𝑓) | |
7 | 5, 6 | mpdan 675 | 1 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∃𝑓 ∈ (UFil‘𝑋)𝐹 ⊆ 𝑓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2051 ∃wrex 3082 Vcvv 3408 ⊆ wss 3822 𝒫 cpw 4416 dom cdm 5403 ‘cfv 6185 cardccrd 9156 Filcfil 22172 UFilcufil 22226 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-ac2 9681 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-se 5363 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-isom 6194 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-rpss 7265 df-om 7395 df-1st 7499 df-2nd 7500 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-oadd 7907 df-er 8087 df-en 8305 df-dom 8306 df-fin 8308 df-fi 8668 df-dju 9122 df-card 9160 df-ac 9334 df-fbas 20259 df-fg 20260 df-fil 22173 df-ufil 22228 |
This theorem is referenced by: ufileu 22246 filufint 22247 ufinffr 22256 ufilen 22257 |
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