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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 10328.) (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 23113 | . . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹) | |
2 | pwexg 5322 | . . 3 ⊢ (𝑋 ∈ 𝐹 → 𝒫 𝑋 ∈ V) | |
3 | pwexg 5322 | . . 3 ⊢ (𝒫 𝑋 ∈ V → 𝒫 𝒫 𝑋 ∈ V) | |
4 | numth3 10328 | . . 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 23169 | . 2 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → ∃𝑓 ∈ (UFil‘𝑋)𝐹 ⊆ 𝑓) | |
7 | 5, 6 | mpdan 684 | 1 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∃𝑓 ∈ (UFil‘𝑋)𝐹 ⊆ 𝑓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ∃wrex 3070 Vcvv 3441 ⊆ wss 3898 𝒫 cpw 4548 dom cdm 5621 ‘cfv 6480 cardccrd 9793 Filcfil 23103 UFilcufil 23157 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5230 ax-sep 5244 ax-nul 5251 ax-pow 5309 ax-pr 5373 ax-un 7651 ax-ac2 10321 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4271 df-if 4475 df-pw 4550 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4854 df-int 4896 df-iun 4944 df-br 5094 df-opab 5156 df-mpt 5177 df-tr 5211 df-id 5519 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5576 df-se 5577 df-we 5578 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6239 df-ord 6306 df-on 6307 df-lim 6308 df-suc 6309 df-iota 6432 df-fun 6482 df-fn 6483 df-f 6484 df-f1 6485 df-fo 6486 df-f1o 6487 df-fv 6488 df-isom 6489 df-riota 7294 df-ov 7341 df-oprab 7342 df-mpo 7343 df-rpss 7639 df-om 7782 df-1st 7900 df-2nd 7901 df-frecs 8168 df-wrecs 8199 df-recs 8273 df-rdg 8312 df-1o 8368 df-oadd 8372 df-er 8570 df-en 8806 df-dom 8807 df-fin 8809 df-fi 9269 df-dju 9759 df-card 9797 df-ac 9974 df-fbas 20701 df-fg 20702 df-fil 23104 df-ufil 23159 |
This theorem is referenced by: ufileu 23177 filufint 23178 ufinffr 23187 ufilen 23188 |
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