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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 3478 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | acacni 10137 | . . . . 5 ⊢ ((CHOICE ∧ 𝑥 ∈ V) → AC 𝑥 = V) | |
3 | 2 | elvd 3481 | . . . 4 ⊢ (CHOICE → AC 𝑥 = V) |
4 | 1, 3 | eleqtrrid 2840 | . . 3 ⊢ (CHOICE → 𝑥 ∈ AC 𝑥) |
5 | 4 | alrimiv 1930 | . 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 10040 | . . . . . . . . 9 ⊢ (𝑥 = 𝒫 𝑧 → AC 𝑥 = AC 𝒫 𝑧) | |
9 | 7, 8 | eleq12d 2827 | . . . . . . . 8 ⊢ (𝑥 = 𝒫 𝑧 → (𝑥 ∈ AC 𝑥 ↔ 𝒫 𝑧 ∈ AC 𝒫 𝑧)) |
10 | 6, 9 | spcv 3595 | . . . . . . 7 ⊢ (∀𝑥 𝑥 ∈ AC 𝑥 → 𝒫 𝑧 ∈ AC 𝒫 𝑧) |
11 | vex 3478 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
12 | vex 3478 | . . . . . . . . . . 11 ⊢ 𝑧 ∈ V | |
13 | 12 | canth2 9132 | . . . . . . . . . 10 ⊢ 𝑧 ≺ 𝒫 𝑧 |
14 | sdomdom 8978 | . . . . . . . . . 10 ⊢ (𝑧 ≺ 𝒫 𝑧 → 𝑧 ≼ 𝒫 𝑧) | |
15 | acndom2 10051 | . . . . . . . . . 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 10049 | . . . . . . . . 9 ⊢ (𝑧 ∈ AC 𝒫 𝑧 ↔ 𝑧 ∈ dom card) | |
18 | 16, 17 | sylib 217 | . . . . . . . 8 ⊢ (𝒫 𝑧 ∈ AC 𝒫 𝑧 → 𝑧 ∈ dom card) |
19 | numacn 10046 | . . . . . . . 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 264 | . . . . 5 ⊢ (∀𝑥 𝑥 ∈ AC 𝑥 → (𝑧 ∈ AC 𝑦 ↔ 𝑧 ∈ V)) |
24 | 23 | eqrdv 2730 | . . . 4 ⊢ (∀𝑥 𝑥 ∈ AC 𝑥 → AC 𝑦 = V) |
25 | 24 | alrimiv 1930 | . . 3 ⊢ (∀𝑥 𝑥 ∈ AC 𝑥 → ∀𝑦AC 𝑦 = V) |
26 | dfacacn 10138 | . . 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 1539 = wceq 1541 ∈ wcel 2106 Vcvv 3474 𝒫 cpw 4602 class class class wbr 5148 dom cdm 5676 ≼ cdom 8939 ≺ csdm 8940 cardccrd 9932 AC wacn 9935 CHOICEwac 10112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 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 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-acn 9939 df-ac 10113 |
This theorem is referenced by: (None) |
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