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Mirrors > Home > MPE Home > Th. List > numufl | Structured version Visualization version GIF version |
Description: Consequence of filssufilg 22092: a set whose double powerset is well-orderable satisfies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.) |
Ref | Expression |
---|---|
numufl | ⊢ (𝒫 𝒫 𝑋 ∈ dom card → 𝑋 ∈ UFL) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | filssufilg 22092 | . . . 4 ⊢ ((𝑓 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → ∃𝑔 ∈ (UFil‘𝑋)𝑓 ⊆ 𝑔) | |
2 | 1 | ancoms 452 | . . 3 ⊢ ((𝒫 𝒫 𝑋 ∈ dom card ∧ 𝑓 ∈ (Fil‘𝑋)) → ∃𝑔 ∈ (UFil‘𝑋)𝑓 ⊆ 𝑔) |
3 | 2 | ralrimiva 3175 | . 2 ⊢ (𝒫 𝒫 𝑋 ∈ dom card → ∀𝑓 ∈ (Fil‘𝑋)∃𝑔 ∈ (UFil‘𝑋)𝑓 ⊆ 𝑔) |
4 | pwexr 7239 | . . . 4 ⊢ (𝒫 𝒫 𝑋 ∈ dom card → 𝒫 𝑋 ∈ V) | |
5 | pwexb 7240 | . . . 4 ⊢ (𝑋 ∈ V ↔ 𝒫 𝑋 ∈ V) | |
6 | 4, 5 | sylibr 226 | . . 3 ⊢ (𝒫 𝒫 𝑋 ∈ dom card → 𝑋 ∈ V) |
7 | isufl 22094 | . . 3 ⊢ (𝑋 ∈ V → (𝑋 ∈ UFL ↔ ∀𝑓 ∈ (Fil‘𝑋)∃𝑔 ∈ (UFil‘𝑋)𝑓 ⊆ 𝑔)) | |
8 | 6, 7 | syl 17 | . 2 ⊢ (𝒫 𝒫 𝑋 ∈ dom card → (𝑋 ∈ UFL ↔ ∀𝑓 ∈ (Fil‘𝑋)∃𝑔 ∈ (UFil‘𝑋)𝑓 ⊆ 𝑔)) |
9 | 3, 8 | mpbird 249 | 1 ⊢ (𝒫 𝒫 𝑋 ∈ dom card → 𝑋 ∈ UFL) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∈ wcel 2164 ∀wral 3117 ∃wrex 3118 Vcvv 3414 ⊆ wss 3798 𝒫 cpw 4380 dom cdm 5346 ‘cfv 6127 cardccrd 9081 Filcfil 22026 UFilcufil 22080 UFLcufl 22081 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-se 5306 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-isom 6136 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-rpss 7202 df-om 7332 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-en 8229 df-dom 8230 df-fin 8232 df-fi 8592 df-card 9085 df-cda 9312 df-fbas 20110 df-fg 20111 df-fil 22027 df-ufil 22082 df-ufl 22083 |
This theorem is referenced by: fiufl 22097 acufl 22098 |
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