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Theorem grothac 9508
Description: The Tarski-Grothendieck Axiom implies the Axiom of Choice (in the form of cardeqv 9151). This can be put in a more conventional form via ween 8718 and dfac8 8817. 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 4110 . . . . . . . . . 10 (𝑥 = 𝑦 → 𝒫 𝑥 = 𝒫 𝑦)
21sseq1d 3594 . . . . . . . . 9 (𝑥 = 𝑦 → (𝒫 𝑥𝑢 ↔ 𝒫 𝑦𝑢))
31eleq1d 2671 . . . . . . . . 9 (𝑥 = 𝑦 → (𝒫 𝑥𝑢 ↔ 𝒫 𝑦𝑢))
42, 3anbi12d 742 . . . . . . . 8 (𝑥 = 𝑦 → ((𝒫 𝑥𝑢 ∧ 𝒫 𝑥𝑢) ↔ (𝒫 𝑦𝑢 ∧ 𝒫 𝑦𝑢)))
54rspcva 3279 . . . . . . 7 ((𝑦𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ 𝒫 𝑥𝑢)) → (𝒫 𝑦𝑢 ∧ 𝒫 𝑦𝑢))
65simpld 473 . . . . . 6 ((𝑦𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ 𝒫 𝑥𝑢)) → 𝒫 𝑦𝑢)
7 rabss 3641 . . . . . . 7 ({𝑥 ∈ 𝒫 𝑢𝑥𝑢} ⊆ 𝑢 ↔ ∀𝑥 ∈ 𝒫 𝑢(𝑥𝑢𝑥𝑢))
87biimpri 216 . . . . . 6 (∀𝑥 ∈ 𝒫 𝑢(𝑥𝑢𝑥𝑢) → {𝑥 ∈ 𝒫 𝑢𝑥𝑢} ⊆ 𝑢)
9 vex 3175 . . . . . . . . . 10 𝑦 ∈ V
109canth2 7975 . . . . . . . . 9 𝑦 ≺ 𝒫 𝑦
11 sdomdom 7846 . . . . . . . . 9 (𝑦 ≺ 𝒫 𝑦𝑦 ≼ 𝒫 𝑦)
1210, 11ax-mp 5 . . . . . . . 8 𝑦 ≼ 𝒫 𝑦
13 vex 3175 . . . . . . . . 9 𝑢 ∈ V
14 ssdomg 7864 . . . . . . . . 9 (𝑢 ∈ V → (𝒫 𝑦𝑢 → 𝒫 𝑦𝑢))
1513, 14ax-mp 5 . . . . . . . 8 (𝒫 𝑦𝑢 → 𝒫 𝑦𝑢)
16 domtr 7872 . . . . . . . 8 ((𝑦 ≼ 𝒫 𝑦 ∧ 𝒫 𝑦𝑢) → 𝑦𝑢)
1712, 15, 16sylancr 693 . . . . . . 7 (𝒫 𝑦𝑢𝑦𝑢)
18 tskwe 8636 . . . . . . . 8 ((𝑢 ∈ V ∧ {𝑥 ∈ 𝒫 𝑢𝑥𝑢} ⊆ 𝑢) → 𝑢 ∈ dom card)
1913, 18mpan 701 . . . . . . 7 ({𝑥 ∈ 𝒫 𝑢𝑥𝑢} ⊆ 𝑢𝑢 ∈ dom card)
20 numdom 8721 . . . . . . . 8 ((𝑢 ∈ dom card ∧ 𝑦𝑢) → 𝑦 ∈ dom card)
2120expcom 449 . . . . . . 7 (𝑦𝑢 → (𝑢 ∈ dom card → 𝑦 ∈ dom card))
2217, 19, 21syl2im 39 . . . . . 6 (𝒫 𝑦𝑢 → ({𝑥 ∈ 𝒫 𝑢𝑥𝑢} ⊆ 𝑢𝑦 ∈ dom card))
236, 8, 22syl2im 39 . . . . 5 ((𝑦𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ 𝒫 𝑥𝑢)) → (∀𝑥 ∈ 𝒫 𝑢(𝑥𝑢𝑥𝑢) → 𝑦 ∈ dom card))
24233impia 1252 . . . 4 ((𝑦𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ 𝒫 𝑥𝑢) ∧ ∀𝑥 ∈ 𝒫 𝑢(𝑥𝑢𝑥𝑢)) → 𝑦 ∈ dom card)
25 axgroth6 9506 . . . 4 𝑢(𝑦𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ 𝒫 𝑥𝑢) ∧ ∀𝑥 ∈ 𝒫 𝑢(𝑥𝑢𝑥𝑢))
2624, 25exlimiiv 1845 . . 3 𝑦 ∈ dom card
2726, 92th 252 . 2 (𝑦 ∈ dom card ↔ 𝑦 ∈ V)
2827eqriv 2606 1 dom card = V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1976  wral 2895  {crab 2899  Vcvv 3172  wss 3539  𝒫 cpw 4107   class class class wbr 4577  dom cdm 5028  cdom 7816  csdm 7817  cardccrd 8621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-groth 9501
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-wrecs 7271  df-recs 7332  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-card 8625
This theorem is referenced by:  axgroth3  9509
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