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Theorem grothac 9855
Description: The Tarski-Grothendieck Axiom implies the Axiom of Choice (in the form of cardeqv 9494). This can be put in a more conventional form via ween 9059 and dfac8 9160. 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.)
Assertion
Ref Expression
grothac dom card = V

Proof of Theorem grothac
Dummy variables 𝑥 𝑦 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pweq 4301 . . . . . . . . . 10 (𝑥 = 𝑦 → 𝒫 𝑥 = 𝒫 𝑦)
21sseq1d 3782 . . . . . . . . 9 (𝑥 = 𝑦 → (𝒫 𝑥𝑢 ↔ 𝒫 𝑦𝑢))
31eleq1d 2835 . . . . . . . . 9 (𝑥 = 𝑦 → (𝒫 𝑥𝑢 ↔ 𝒫 𝑦𝑢))
42, 3anbi12d 610 . . . . . . . 8 (𝑥 = 𝑦 → ((𝒫 𝑥𝑢 ∧ 𝒫 𝑥𝑢) ↔ (𝒫 𝑦𝑢 ∧ 𝒫 𝑦𝑢)))
54rspcva 3459 . . . . . . 7 ((𝑦𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ 𝒫 𝑥𝑢)) → (𝒫 𝑦𝑢 ∧ 𝒫 𝑦𝑢))
65simpld 478 . . . . . 6 ((𝑦𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ 𝒫 𝑥𝑢)) → 𝒫 𝑦𝑢)
7 rabss 3829 . . . . . . 7 ({𝑥 ∈ 𝒫 𝑢𝑥𝑢} ⊆ 𝑢 ↔ ∀𝑥 ∈ 𝒫 𝑢(𝑥𝑢𝑥𝑢))
87biimpri 218 . . . . . 6 (∀𝑥 ∈ 𝒫 𝑢(𝑥𝑢𝑥𝑢) → {𝑥 ∈ 𝒫 𝑢𝑥𝑢} ⊆ 𝑢)
9 vex 3354 . . . . . . . . . 10 𝑦 ∈ V
109canth2 8270 . . . . . . . . 9 𝑦 ≺ 𝒫 𝑦
11 sdomdom 8138 . . . . . . . . 9 (𝑦 ≺ 𝒫 𝑦𝑦 ≼ 𝒫 𝑦)
1210, 11ax-mp 5 . . . . . . . 8 𝑦 ≼ 𝒫 𝑦
13 vex 3354 . . . . . . . . 9 𝑢 ∈ V
14 ssdomg 8156 . . . . . . . . 9 (𝑢 ∈ V → (𝒫 𝑦𝑢 → 𝒫 𝑦𝑢))
1513, 14ax-mp 5 . . . . . . . 8 (𝒫 𝑦𝑢 → 𝒫 𝑦𝑢)
16 domtr 8163 . . . . . . . 8 ((𝑦 ≼ 𝒫 𝑦 ∧ 𝒫 𝑦𝑢) → 𝑦𝑢)
1712, 15, 16sylancr 569 . . . . . . 7 (𝒫 𝑦𝑢𝑦𝑢)
18 tskwe 8977 . . . . . . . 8 ((𝑢 ∈ V ∧ {𝑥 ∈ 𝒫 𝑢𝑥𝑢} ⊆ 𝑢) → 𝑢 ∈ dom card)
1913, 18mpan 664 . . . . . . 7 ({𝑥 ∈ 𝒫 𝑢𝑥𝑢} ⊆ 𝑢𝑢 ∈ dom card)
20 numdom 9062 . . . . . . . 8 ((𝑢 ∈ dom card ∧ 𝑦𝑢) → 𝑦 ∈ dom card)
2120expcom 398 . . . . . . 7 (𝑦𝑢 → (𝑢 ∈ dom card → 𝑦 ∈ dom card))
2217, 19, 21syl2im 40 . . . . . 6 (𝒫 𝑦𝑢 → ({𝑥 ∈ 𝒫 𝑢𝑥𝑢} ⊆ 𝑢𝑦 ∈ dom card))
236, 8, 22syl2im 40 . . . . 5 ((𝑦𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ 𝒫 𝑥𝑢)) → (∀𝑥 ∈ 𝒫 𝑢(𝑥𝑢𝑥𝑢) → 𝑦 ∈ dom card))
24233impia 1109 . . . 4 ((𝑦𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ 𝒫 𝑥𝑢) ∧ ∀𝑥 ∈ 𝒫 𝑢(𝑥𝑢𝑥𝑢)) → 𝑦 ∈ dom card)
25 axgroth6 9853 . . . 4 𝑢(𝑦𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ 𝒫 𝑥𝑢) ∧ ∀𝑥 ∈ 𝒫 𝑢(𝑥𝑢𝑥𝑢))
2624, 25exlimiiv 2011 . . 3 𝑦 ∈ dom card
2726, 92th 254 . 2 (𝑦 ∈ dom card ↔ 𝑦 ∈ V)
2827eqriv 2768 1 dom card = V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  {crab 3065  Vcvv 3351  wss 3724  𝒫 cpw 4298   class class class wbr 4787  dom cdm 5250  cdom 8108  csdm 8109  cardccrd 8962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097  ax-groth 9848
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-pss 3740  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-se 5210  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5824  df-ord 5870  df-on 5871  df-suc 5873  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-isom 6041  df-riota 6755  df-wrecs 7560  df-recs 7622  df-er 7897  df-en 8111  df-dom 8112  df-sdom 8113  df-card 8966
This theorem is referenced by:  axgroth3  9856
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