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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 3465 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
2 | 1 | pwex 5380 | . . . . . 6 ⊢ 𝒫 𝑥 ∈ V |
3 | 2 | pwex 5380 | . . . . 5 ⊢ 𝒫 𝒫 𝑥 ∈ V |
4 | dfac10 10162 | . . . . . 6 ⊢ (CHOICE ↔ dom card = V) | |
5 | 4 | biimpi 215 | . . . . 5 ⊢ (CHOICE → dom card = V) |
6 | 3, 5 | eleqtrrid 2832 | . . . 4 ⊢ (CHOICE → 𝒫 𝒫 𝑥 ∈ dom card) |
7 | numufl 23863 | . . . 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 264 | . 2 ⊢ (CHOICE → (𝑥 ∈ UFL ↔ 𝑥 ∈ V)) |
11 | 10 | eqrdv 2723 | 1 ⊢ (CHOICE → UFL = V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3461 𝒫 cpw 4604 dom cdm 5678 cardccrd 9960 CHOICEwac 10140 UFLcufl 23848 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-rpss 7729 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-oadd 8491 df-er 8725 df-en 8965 df-dom 8966 df-fin 8968 df-fi 9436 df-dju 9926 df-card 9964 df-ac 10141 df-fbas 21293 df-fg 21294 df-fil 23794 df-ufil 23849 df-ufl 23850 |
This theorem is referenced by: ptcmp 24006 dfac21 42629 |
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