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Mirrors > Home > MPE Home > Th. List > fiufl | Structured version Visualization version GIF version |
Description: A finite set satisfies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.) |
Ref | Expression |
---|---|
fiufl | ⊢ (𝑋 ∈ Fin → 𝑋 ∈ UFL) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwfi 8811 | . 2 ⊢ (𝑋 ∈ Fin ↔ 𝒫 𝑋 ∈ Fin) | |
2 | pwfi 8811 | . . 3 ⊢ (𝒫 𝑋 ∈ Fin ↔ 𝒫 𝒫 𝑋 ∈ Fin) | |
3 | finnum 9369 | . . . 4 ⊢ (𝒫 𝒫 𝑋 ∈ Fin → 𝒫 𝒫 𝑋 ∈ dom card) | |
4 | numufl 22515 | . . . 4 ⊢ (𝒫 𝒫 𝑋 ∈ dom card → 𝑋 ∈ UFL) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝒫 𝒫 𝑋 ∈ Fin → 𝑋 ∈ UFL) |
6 | 2, 5 | sylbi 219 | . 2 ⊢ (𝒫 𝑋 ∈ Fin → 𝑋 ∈ UFL) |
7 | 1, 6 | sylbi 219 | 1 ⊢ (𝑋 ∈ Fin → 𝑋 ∈ UFL) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2108 𝒫 cpw 4537 dom cdm 5548 Fincfn 8501 cardccrd 9356 UFLcufl 22500 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-rpss 7441 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-2o 8095 df-oadd 8098 df-er 8281 df-map 8400 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-fi 8867 df-dju 9322 df-card 9360 df-fbas 20534 df-fg 20535 df-fil 22446 df-ufil 22501 df-ufl 22502 |
This theorem is referenced by: (None) |
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