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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 3497 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | acacni 9566 | . . . . 5 ⊢ ((CHOICE ∧ 𝑥 ∈ V) → AC 𝑥 = V) | |
3 | 2 | elvd 3500 | . . . 4 ⊢ (CHOICE → AC 𝑥 = V) |
4 | 1, 3 | eleqtrrid 2920 | . . 3 ⊢ (CHOICE → 𝑥 ∈ AC 𝑥) |
5 | 4 | alrimiv 1928 | . 2 ⊢ (CHOICE → ∀𝑥 𝑥 ∈ AC 𝑥) |
6 | vpwex 5278 | . . . . . . . 8 ⊢ 𝒫 𝑧 ∈ V | |
7 | id 22 | . . . . . . . . 9 ⊢ (𝑥 = 𝒫 𝑧 → 𝑥 = 𝒫 𝑧) | |
8 | acneq 9469 | . . . . . . . . 9 ⊢ (𝑥 = 𝒫 𝑧 → AC 𝑥 = AC 𝒫 𝑧) | |
9 | 7, 8 | eleq12d 2907 | . . . . . . . 8 ⊢ (𝑥 = 𝒫 𝑧 → (𝑥 ∈ AC 𝑥 ↔ 𝒫 𝑧 ∈ AC 𝒫 𝑧)) |
10 | 6, 9 | spcv 3606 | . . . . . . 7 ⊢ (∀𝑥 𝑥 ∈ AC 𝑥 → 𝒫 𝑧 ∈ AC 𝒫 𝑧) |
11 | vex 3497 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
12 | vex 3497 | . . . . . . . . . . 11 ⊢ 𝑧 ∈ V | |
13 | 12 | canth2 8670 | . . . . . . . . . 10 ⊢ 𝑧 ≺ 𝒫 𝑧 |
14 | sdomdom 8537 | . . . . . . . . . 10 ⊢ (𝑧 ≺ 𝒫 𝑧 → 𝑧 ≼ 𝒫 𝑧) | |
15 | acndom2 9480 | . . . . . . . . . 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 9478 | . . . . . . . . 9 ⊢ (𝑧 ∈ AC 𝒫 𝑧 ↔ 𝑧 ∈ dom card) | |
18 | 16, 17 | sylib 220 | . . . . . . . 8 ⊢ (𝒫 𝑧 ∈ AC 𝒫 𝑧 → 𝑧 ∈ dom card) |
19 | numacn 9475 | . . . . . . . 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 267 | . . . . 5 ⊢ (∀𝑥 𝑥 ∈ AC 𝑥 → (𝑧 ∈ AC 𝑦 ↔ 𝑧 ∈ V)) |
24 | 23 | eqrdv 2819 | . . . 4 ⊢ (∀𝑥 𝑥 ∈ AC 𝑥 → AC 𝑦 = V) |
25 | 24 | alrimiv 1928 | . . 3 ⊢ (∀𝑥 𝑥 ∈ AC 𝑥 → ∀𝑦AC 𝑦 = V) |
26 | dfacacn 9567 | . . 3 ⊢ (CHOICE ↔ ∀𝑦AC 𝑦 = V) | |
27 | 25, 26 | sylibr 236 | . 2 ⊢ (∀𝑥 𝑥 ∈ AC 𝑥 → CHOICE) |
28 | 5, 27 | impbii 211 | 1 ⊢ (CHOICE ↔ ∀𝑥 𝑥 ∈ AC 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1535 = wceq 1537 ∈ wcel 2114 Vcvv 3494 𝒫 cpw 4539 class class class wbr 5066 dom cdm 5555 ≼ cdom 8507 ≺ csdm 8508 cardccrd 9364 AC wacn 9367 CHOICEwac 9541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-1o 8102 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-card 9368 df-acn 9371 df-ac 9542 |
This theorem is referenced by: (None) |
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