![]() |
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 3482 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | acacni 10179 | . . . . 5 ⊢ ((CHOICE ∧ 𝑥 ∈ V) → AC 𝑥 = V) | |
3 | 2 | elvd 3484 | . . . 4 ⊢ (CHOICE → AC 𝑥 = V) |
4 | 1, 3 | eleqtrrid 2846 | . . 3 ⊢ (CHOICE → 𝑥 ∈ AC 𝑥) |
5 | 4 | alrimiv 1925 | . 2 ⊢ (CHOICE → ∀𝑥 𝑥 ∈ AC 𝑥) |
6 | vpwex 5383 | . . . . . . . 8 ⊢ 𝒫 𝑧 ∈ V | |
7 | id 22 | . . . . . . . . 9 ⊢ (𝑥 = 𝒫 𝑧 → 𝑥 = 𝒫 𝑧) | |
8 | acneq 10081 | . . . . . . . . 9 ⊢ (𝑥 = 𝒫 𝑧 → AC 𝑥 = AC 𝒫 𝑧) | |
9 | 7, 8 | eleq12d 2833 | . . . . . . . 8 ⊢ (𝑥 = 𝒫 𝑧 → (𝑥 ∈ AC 𝑥 ↔ 𝒫 𝑧 ∈ AC 𝒫 𝑧)) |
10 | 6, 9 | spcv 3605 | . . . . . . 7 ⊢ (∀𝑥 𝑥 ∈ AC 𝑥 → 𝒫 𝑧 ∈ AC 𝒫 𝑧) |
11 | vex 3482 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
12 | vex 3482 | . . . . . . . . . . 11 ⊢ 𝑧 ∈ V | |
13 | 12 | canth2 9169 | . . . . . . . . . 10 ⊢ 𝑧 ≺ 𝒫 𝑧 |
14 | sdomdom 9019 | . . . . . . . . . 10 ⊢ (𝑧 ≺ 𝒫 𝑧 → 𝑧 ≼ 𝒫 𝑧) | |
15 | acndom2 10092 | . . . . . . . . . 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 10090 | . . . . . . . . 9 ⊢ (𝑧 ∈ AC 𝒫 𝑧 ↔ 𝑧 ∈ dom card) | |
18 | 16, 17 | sylib 218 | . . . . . . . 8 ⊢ (𝒫 𝑧 ∈ AC 𝒫 𝑧 → 𝑧 ∈ dom card) |
19 | numacn 10087 | . . . . . . . 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 2733 | . . . 4 ⊢ (∀𝑥 𝑥 ∈ AC 𝑥 → AC 𝑦 = V) |
25 | 24 | alrimiv 1925 | . . 3 ⊢ (∀𝑥 𝑥 ∈ AC 𝑥 → ∀𝑦AC 𝑦 = V) |
26 | dfacacn 10180 | . . 3 ⊢ (CHOICE ↔ ∀𝑦AC 𝑦 = V) | |
27 | 25, 26 | sylibr 234 | . 2 ⊢ (∀𝑥 𝑥 ∈ AC 𝑥 → CHOICE) |
28 | 5, 27 | impbii 209 | 1 ⊢ (CHOICE ↔ ∀𝑥 𝑥 ∈ AC 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∀wal 1535 = wceq 1537 ∈ wcel 2106 Vcvv 3478 𝒫 cpw 4605 class class class wbr 5148 dom cdm 5689 ≼ cdom 8982 ≺ csdm 8983 cardccrd 9973 AC wacn 9976 CHOICEwac 10153 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-acn 9980 df-ac 10154 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |