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Mirrors > Home > MPE Home > Th. List > acufl | Structured version Visualization version GIF version |
Description: The axiom of choice implies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.) |
Ref | Expression |
---|---|
acufl | ⊢ (CHOICE → UFL = V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3416 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
2 | 1 | pwex 5079 | . . . . . 6 ⊢ 𝒫 𝑥 ∈ V |
3 | 2 | pwex 5079 | . . . . 5 ⊢ 𝒫 𝒫 𝑥 ∈ V |
4 | dfac10 9273 | . . . . . 6 ⊢ (CHOICE ↔ dom card = V) | |
5 | 4 | biimpi 208 | . . . . 5 ⊢ (CHOICE → dom card = V) |
6 | 3, 5 | syl5eleqr 2912 | . . . 4 ⊢ (CHOICE → 𝒫 𝒫 𝑥 ∈ dom card) |
7 | numufl 22088 | . . . 4 ⊢ (𝒫 𝒫 𝑥 ∈ dom card → 𝑥 ∈ UFL) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (CHOICE → 𝑥 ∈ UFL) |
9 | 1 | a1i 11 | . . 3 ⊢ (CHOICE → 𝑥 ∈ V) |
10 | 8, 9 | 2thd 257 | . 2 ⊢ (CHOICE → (𝑥 ∈ UFL ↔ 𝑥 ∈ V)) |
11 | 10 | eqrdv 2822 | 1 ⊢ (CHOICE → UFL = V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 Vcvv 3413 𝒫 cpw 4377 dom cdm 5341 cardccrd 9073 CHOICEwac 9250 UFLcufl 22073 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-int 4697 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-se 5301 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-isom 6131 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-rpss 7196 df-om 7326 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-1o 7825 df-oadd 7829 df-er 8008 df-en 8222 df-dom 8223 df-fin 8225 df-fi 8585 df-card 9077 df-ac 9251 df-cda 9304 df-fbas 20102 df-fg 20103 df-fil 22019 df-ufil 22074 df-ufl 22075 |
This theorem is referenced by: ptcmp 22231 dfac21 38478 |
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