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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 3176 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | acacni 8845 | . . . . 5 ⊢ ((CHOICE ∧ 𝑥 ∈ V) → AC 𝑥 = V) | |
3 | 1, 2 | mpan2 703 | . . . 4 ⊢ (CHOICE → AC 𝑥 = V) |
4 | 1, 3 | syl5eleqr 2695 | . . 3 ⊢ (CHOICE → 𝑥 ∈ AC 𝑥) |
5 | 4 | alrimiv 1842 | . 2 ⊢ (CHOICE → ∀𝑥 𝑥 ∈ AC 𝑥) |
6 | vpwex 4775 | . . . . . . . 8 ⊢ 𝒫 𝑧 ∈ V | |
7 | id 22 | . . . . . . . . 9 ⊢ (𝑥 = 𝒫 𝑧 → 𝑥 = 𝒫 𝑧) | |
8 | acneq 8749 | . . . . . . . . 9 ⊢ (𝑥 = 𝒫 𝑧 → AC 𝑥 = AC 𝒫 𝑧) | |
9 | 7, 8 | eleq12d 2682 | . . . . . . . 8 ⊢ (𝑥 = 𝒫 𝑧 → (𝑥 ∈ AC 𝑥 ↔ 𝒫 𝑧 ∈ AC 𝒫 𝑧)) |
10 | 6, 9 | spcv 3272 | . . . . . . 7 ⊢ (∀𝑥 𝑥 ∈ AC 𝑥 → 𝒫 𝑧 ∈ AC 𝒫 𝑧) |
11 | vex 3176 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
12 | vex 3176 | . . . . . . . . . . 11 ⊢ 𝑧 ∈ V | |
13 | 12 | canth2 7998 | . . . . . . . . . 10 ⊢ 𝑧 ≺ 𝒫 𝑧 |
14 | sdomdom 7869 | . . . . . . . . . 10 ⊢ (𝑧 ≺ 𝒫 𝑧 → 𝑧 ≼ 𝒫 𝑧) | |
15 | acndom2 8760 | . . . . . . . . . 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 8758 | . . . . . . . . 9 ⊢ (𝑧 ∈ AC 𝒫 𝑧 ↔ 𝑧 ∈ dom card) | |
18 | 16, 17 | sylib 207 | . . . . . . . 8 ⊢ (𝒫 𝑧 ∈ AC 𝒫 𝑧 → 𝑧 ∈ dom card) |
19 | numacn 8755 | . . . . . . . 8 ⊢ (𝑦 ∈ V → (𝑧 ∈ dom card → 𝑧 ∈ AC 𝑦)) | |
20 | 11, 18, 19 | mpsyl 66 | . . . . . . 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 254 | . . . . 5 ⊢ (∀𝑥 𝑥 ∈ AC 𝑥 → (𝑧 ∈ AC 𝑦 ↔ 𝑧 ∈ V)) |
24 | 23 | eqrdv 2608 | . . . 4 ⊢ (∀𝑥 𝑥 ∈ AC 𝑥 → AC 𝑦 = V) |
25 | 24 | alrimiv 1842 | . . 3 ⊢ (∀𝑥 𝑥 ∈ AC 𝑥 → ∀𝑦AC 𝑦 = V) |
26 | dfacacn 8846 | . . 3 ⊢ (CHOICE ↔ ∀𝑦AC 𝑦 = V) | |
27 | 25, 26 | sylibr 223 | . 2 ⊢ (∀𝑥 𝑥 ∈ AC 𝑥 → CHOICE) |
28 | 5, 27 | impbii 198 | 1 ⊢ (CHOICE ↔ ∀𝑥 𝑥 ∈ AC 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 = wceq 1475 ∈ wcel 1977 Vcvv 3173 𝒫 cpw 4108 class class class wbr 4583 dom cdm 5038 ≼ cdom 7839 ≺ csdm 7840 cardccrd 8644 AC wacn 8647 CHOICEwac 8821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-1o 7447 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-acn 8651 df-ac 8822 |
This theorem is referenced by: (None) |
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