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Mirrors > Home > MPE Home > Th. List > grothac | Structured version Visualization version GIF version |
Description: The Tarski-Grothendieck Axiom implies the Axiom of Choice (in the form of cardeqv 9329). This can be put in a more conventional form via ween 8896 and dfac8 8995. Note that the mere existence of strongly inaccessible cardinals doesn't imply AC, but rather the particular form of the Tarski-Grothendieck axiom (see http://www.cs.nyu.edu/pipermail/fom/2008-March/012783.html). (Contributed by Mario Carneiro, 19-Apr-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
grothac | ⊢ dom card = V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pweq 4194 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝒫 𝑥 = 𝒫 𝑦) | |
2 | 1 | sseq1d 3665 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝒫 𝑥 ⊆ 𝑢 ↔ 𝒫 𝑦 ⊆ 𝑢)) |
3 | 1 | eleq1d 2715 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝒫 𝑥 ∈ 𝑢 ↔ 𝒫 𝑦 ∈ 𝑢)) |
4 | 2, 3 | anbi12d 747 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝒫 𝑥 ⊆ 𝑢 ∧ 𝒫 𝑥 ∈ 𝑢) ↔ (𝒫 𝑦 ⊆ 𝑢 ∧ 𝒫 𝑦 ∈ 𝑢))) |
5 | 4 | rspcva 3338 | . . . . . . 7 ⊢ ((𝑦 ∈ 𝑢 ∧ ∀𝑥 ∈ 𝑢 (𝒫 𝑥 ⊆ 𝑢 ∧ 𝒫 𝑥 ∈ 𝑢)) → (𝒫 𝑦 ⊆ 𝑢 ∧ 𝒫 𝑦 ∈ 𝑢)) |
6 | 5 | simpld 474 | . . . . . 6 ⊢ ((𝑦 ∈ 𝑢 ∧ ∀𝑥 ∈ 𝑢 (𝒫 𝑥 ⊆ 𝑢 ∧ 𝒫 𝑥 ∈ 𝑢)) → 𝒫 𝑦 ⊆ 𝑢) |
7 | rabss 3712 | . . . . . . 7 ⊢ ({𝑥 ∈ 𝒫 𝑢 ∣ 𝑥 ≺ 𝑢} ⊆ 𝑢 ↔ ∀𝑥 ∈ 𝒫 𝑢(𝑥 ≺ 𝑢 → 𝑥 ∈ 𝑢)) | |
8 | 7 | biimpri 218 | . . . . . 6 ⊢ (∀𝑥 ∈ 𝒫 𝑢(𝑥 ≺ 𝑢 → 𝑥 ∈ 𝑢) → {𝑥 ∈ 𝒫 𝑢 ∣ 𝑥 ≺ 𝑢} ⊆ 𝑢) |
9 | vex 3234 | . . . . . . . . . 10 ⊢ 𝑦 ∈ V | |
10 | 9 | canth2 8154 | . . . . . . . . 9 ⊢ 𝑦 ≺ 𝒫 𝑦 |
11 | sdomdom 8025 | . . . . . . . . 9 ⊢ (𝑦 ≺ 𝒫 𝑦 → 𝑦 ≼ 𝒫 𝑦) | |
12 | 10, 11 | ax-mp 5 | . . . . . . . 8 ⊢ 𝑦 ≼ 𝒫 𝑦 |
13 | vex 3234 | . . . . . . . . 9 ⊢ 𝑢 ∈ V | |
14 | ssdomg 8043 | . . . . . . . . 9 ⊢ (𝑢 ∈ V → (𝒫 𝑦 ⊆ 𝑢 → 𝒫 𝑦 ≼ 𝑢)) | |
15 | 13, 14 | ax-mp 5 | . . . . . . . 8 ⊢ (𝒫 𝑦 ⊆ 𝑢 → 𝒫 𝑦 ≼ 𝑢) |
16 | domtr 8050 | . . . . . . . 8 ⊢ ((𝑦 ≼ 𝒫 𝑦 ∧ 𝒫 𝑦 ≼ 𝑢) → 𝑦 ≼ 𝑢) | |
17 | 12, 15, 16 | sylancr 696 | . . . . . . 7 ⊢ (𝒫 𝑦 ⊆ 𝑢 → 𝑦 ≼ 𝑢) |
18 | tskwe 8814 | . . . . . . . 8 ⊢ ((𝑢 ∈ V ∧ {𝑥 ∈ 𝒫 𝑢 ∣ 𝑥 ≺ 𝑢} ⊆ 𝑢) → 𝑢 ∈ dom card) | |
19 | 13, 18 | mpan 706 | . . . . . . 7 ⊢ ({𝑥 ∈ 𝒫 𝑢 ∣ 𝑥 ≺ 𝑢} ⊆ 𝑢 → 𝑢 ∈ dom card) |
20 | numdom 8899 | . . . . . . . 8 ⊢ ((𝑢 ∈ dom card ∧ 𝑦 ≼ 𝑢) → 𝑦 ∈ dom card) | |
21 | 20 | expcom 450 | . . . . . . 7 ⊢ (𝑦 ≼ 𝑢 → (𝑢 ∈ dom card → 𝑦 ∈ dom card)) |
22 | 17, 19, 21 | syl2im 40 | . . . . . 6 ⊢ (𝒫 𝑦 ⊆ 𝑢 → ({𝑥 ∈ 𝒫 𝑢 ∣ 𝑥 ≺ 𝑢} ⊆ 𝑢 → 𝑦 ∈ dom card)) |
23 | 6, 8, 22 | syl2im 40 | . . . . 5 ⊢ ((𝑦 ∈ 𝑢 ∧ ∀𝑥 ∈ 𝑢 (𝒫 𝑥 ⊆ 𝑢 ∧ 𝒫 𝑥 ∈ 𝑢)) → (∀𝑥 ∈ 𝒫 𝑢(𝑥 ≺ 𝑢 → 𝑥 ∈ 𝑢) → 𝑦 ∈ dom card)) |
24 | 23 | 3impia 1280 | . . . 4 ⊢ ((𝑦 ∈ 𝑢 ∧ ∀𝑥 ∈ 𝑢 (𝒫 𝑥 ⊆ 𝑢 ∧ 𝒫 𝑥 ∈ 𝑢) ∧ ∀𝑥 ∈ 𝒫 𝑢(𝑥 ≺ 𝑢 → 𝑥 ∈ 𝑢)) → 𝑦 ∈ dom card) |
25 | axgroth6 9688 | . . . 4 ⊢ ∃𝑢(𝑦 ∈ 𝑢 ∧ ∀𝑥 ∈ 𝑢 (𝒫 𝑥 ⊆ 𝑢 ∧ 𝒫 𝑥 ∈ 𝑢) ∧ ∀𝑥 ∈ 𝒫 𝑢(𝑥 ≺ 𝑢 → 𝑥 ∈ 𝑢)) | |
26 | 24, 25 | exlimiiv 1899 | . . 3 ⊢ 𝑦 ∈ dom card |
27 | 26, 9 | 2th 254 | . 2 ⊢ (𝑦 ∈ dom card ↔ 𝑦 ∈ V) |
28 | 27 | eqriv 2648 | 1 ⊢ dom card = V |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ∀wral 2941 {crab 2945 Vcvv 3231 ⊆ wss 3607 𝒫 cpw 4191 class class class wbr 4685 dom cdm 5143 ≼ cdom 7995 ≺ csdm 7996 cardccrd 8799 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-groth 9683 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-wrecs 7452 df-recs 7513 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-card 8803 |
This theorem is referenced by: axgroth3 9691 |
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