![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > dfac13 | Structured version Visualization version GIF version |
Description: The axiom of choice holds iff every set has choice sequences as long as itself. (Contributed by Mario Carneiro, 3-Sep-2015.) |
Ref | Expression |
---|---|
dfac13 | ⊢ (CHOICE ↔ ∀𝑥 𝑥 ∈ AC 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3479 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | acacni 10132 | . . . . 5 ⊢ ((CHOICE ∧ 𝑥 ∈ V) → AC 𝑥 = V) | |
3 | 2 | elvd 3482 | . . . 4 ⊢ (CHOICE → AC 𝑥 = V) |
4 | 1, 3 | eleqtrrid 2841 | . . 3 ⊢ (CHOICE → 𝑥 ∈ AC 𝑥) |
5 | 4 | alrimiv 1931 | . 2 ⊢ (CHOICE → ∀𝑥 𝑥 ∈ AC 𝑥) |
6 | vpwex 5375 | . . . . . . . 8 ⊢ 𝒫 𝑧 ∈ V | |
7 | id 22 | . . . . . . . . 9 ⊢ (𝑥 = 𝒫 𝑧 → 𝑥 = 𝒫 𝑧) | |
8 | acneq 10035 | . . . . . . . . 9 ⊢ (𝑥 = 𝒫 𝑧 → AC 𝑥 = AC 𝒫 𝑧) | |
9 | 7, 8 | eleq12d 2828 | . . . . . . . 8 ⊢ (𝑥 = 𝒫 𝑧 → (𝑥 ∈ AC 𝑥 ↔ 𝒫 𝑧 ∈ AC 𝒫 𝑧)) |
10 | 6, 9 | spcv 3596 | . . . . . . 7 ⊢ (∀𝑥 𝑥 ∈ AC 𝑥 → 𝒫 𝑧 ∈ AC 𝒫 𝑧) |
11 | vex 3479 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
12 | vex 3479 | . . . . . . . . . . 11 ⊢ 𝑧 ∈ V | |
13 | 12 | canth2 9127 | . . . . . . . . . 10 ⊢ 𝑧 ≺ 𝒫 𝑧 |
14 | sdomdom 8973 | . . . . . . . . . 10 ⊢ (𝑧 ≺ 𝒫 𝑧 → 𝑧 ≼ 𝒫 𝑧) | |
15 | acndom2 10046 | . . . . . . . . . 10 ⊢ (𝑧 ≼ 𝒫 𝑧 → (𝒫 𝑧 ∈ AC 𝒫 𝑧 → 𝑧 ∈ AC 𝒫 𝑧)) | |
16 | 13, 14, 15 | mp2b 10 | . . . . . . . . 9 ⊢ (𝒫 𝑧 ∈ AC 𝒫 𝑧 → 𝑧 ∈ AC 𝒫 𝑧) |
17 | acnnum 10044 | . . . . . . . . 9 ⊢ (𝑧 ∈ AC 𝒫 𝑧 ↔ 𝑧 ∈ dom card) | |
18 | 16, 17 | sylib 217 | . . . . . . . 8 ⊢ (𝒫 𝑧 ∈ AC 𝒫 𝑧 → 𝑧 ∈ dom card) |
19 | numacn 10041 | . . . . . . . 8 ⊢ (𝑦 ∈ V → (𝑧 ∈ dom card → 𝑧 ∈ AC 𝑦)) | |
20 | 11, 18, 19 | mpsyl 68 | . . . . . . 7 ⊢ (𝒫 𝑧 ∈ AC 𝒫 𝑧 → 𝑧 ∈ AC 𝑦) |
21 | 10, 20 | syl 17 | . . . . . 6 ⊢ (∀𝑥 𝑥 ∈ AC 𝑥 → 𝑧 ∈ AC 𝑦) |
22 | 12 | a1i 11 | . . . . . 6 ⊢ (∀𝑥 𝑥 ∈ AC 𝑥 → 𝑧 ∈ V) |
23 | 21, 22 | 2thd 265 | . . . . 5 ⊢ (∀𝑥 𝑥 ∈ AC 𝑥 → (𝑧 ∈ AC 𝑦 ↔ 𝑧 ∈ V)) |
24 | 23 | eqrdv 2731 | . . . 4 ⊢ (∀𝑥 𝑥 ∈ AC 𝑥 → AC 𝑦 = V) |
25 | 24 | alrimiv 1931 | . . 3 ⊢ (∀𝑥 𝑥 ∈ AC 𝑥 → ∀𝑦AC 𝑦 = V) |
26 | dfacacn 10133 | . . 3 ⊢ (CHOICE ↔ ∀𝑦AC 𝑦 = V) | |
27 | 25, 26 | sylibr 233 | . 2 ⊢ (∀𝑥 𝑥 ∈ AC 𝑥 → CHOICE) |
28 | 5, 27 | impbii 208 | 1 ⊢ (CHOICE ↔ ∀𝑥 𝑥 ∈ AC 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1540 = wceq 1542 ∈ wcel 2107 Vcvv 3475 𝒫 cpw 4602 class class class wbr 5148 dom cdm 5676 ≼ cdom 8934 ≺ csdm 8935 cardccrd 9927 AC wacn 9930 CHOICEwac 10107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-card 9931 df-acn 9934 df-ac 10108 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |