MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grothac Structured version   Visualization version   GIF version

Theorem grothac 9690
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.)
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 4194 . . . . . . . . . 10 (𝑥 = 𝑦 → 𝒫 𝑥 = 𝒫 𝑦)
21sseq1d 3665 . . . . . . . . 9 (𝑥 = 𝑦 → (𝒫 𝑥𝑢 ↔ 𝒫 𝑦𝑢))
31eleq1d 2715 . . . . . . . . 9 (𝑥 = 𝑦 → (𝒫 𝑥𝑢 ↔ 𝒫 𝑦𝑢))
42, 3anbi12d 747 . . . . . . . 8 (𝑥 = 𝑦 → ((𝒫 𝑥𝑢 ∧ 𝒫 𝑥𝑢) ↔ (𝒫 𝑦𝑢 ∧ 𝒫 𝑦𝑢)))
54rspcva 3338 . . . . . . 7 ((𝑦𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ 𝒫 𝑥𝑢)) → (𝒫 𝑦𝑢 ∧ 𝒫 𝑦𝑢))
65simpld 474 . . . . . 6 ((𝑦𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ 𝒫 𝑥𝑢)) → 𝒫 𝑦𝑢)
7 rabss 3712 . . . . . . 7 ({𝑥 ∈ 𝒫 𝑢𝑥𝑢} ⊆ 𝑢 ↔ ∀𝑥 ∈ 𝒫 𝑢(𝑥𝑢𝑥𝑢))
87biimpri 218 . . . . . 6 (∀𝑥 ∈ 𝒫 𝑢(𝑥𝑢𝑥𝑢) → {𝑥 ∈ 𝒫 𝑢𝑥𝑢} ⊆ 𝑢)
9 vex 3234 . . . . . . . . . 10 𝑦 ∈ V
109canth2 8154 . . . . . . . . 9 𝑦 ≺ 𝒫 𝑦
11 sdomdom 8025 . . . . . . . . 9 (𝑦 ≺ 𝒫 𝑦𝑦 ≼ 𝒫 𝑦)
1210, 11ax-mp 5 . . . . . . . 8 𝑦 ≼ 𝒫 𝑦
13 vex 3234 . . . . . . . . 9 𝑢 ∈ V
14 ssdomg 8043 . . . . . . . . 9 (𝑢 ∈ V → (𝒫 𝑦𝑢 → 𝒫 𝑦𝑢))
1513, 14ax-mp 5 . . . . . . . 8 (𝒫 𝑦𝑢 → 𝒫 𝑦𝑢)
16 domtr 8050 . . . . . . . 8 ((𝑦 ≼ 𝒫 𝑦 ∧ 𝒫 𝑦𝑢) → 𝑦𝑢)
1712, 15, 16sylancr 696 . . . . . . 7 (𝒫 𝑦𝑢𝑦𝑢)
18 tskwe 8814 . . . . . . . 8 ((𝑢 ∈ V ∧ {𝑥 ∈ 𝒫 𝑢𝑥𝑢} ⊆ 𝑢) → 𝑢 ∈ dom card)
1913, 18mpan 706 . . . . . . 7 ({𝑥 ∈ 𝒫 𝑢𝑥𝑢} ⊆ 𝑢𝑢 ∈ dom card)
20 numdom 8899 . . . . . . . 8 ((𝑢 ∈ dom card ∧ 𝑦𝑢) → 𝑦 ∈ dom card)
2120expcom 450 . . . . . . 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 1280 . . . 4 ((𝑦𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ 𝒫 𝑥𝑢) ∧ ∀𝑥 ∈ 𝒫 𝑢(𝑥𝑢𝑥𝑢)) → 𝑦 ∈ dom card)
25 axgroth6 9688 . . . 4 𝑢(𝑦𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ 𝒫 𝑥𝑢) ∧ ∀𝑥 ∈ 𝒫 𝑢(𝑥𝑢𝑥𝑢))
2624, 25exlimiiv 1899 . . 3 𝑦 ∈ dom card
2726, 92th 254 . 2 (𝑦 ∈ dom card ↔ 𝑦 ∈ V)
2827eqriv 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
  Copyright terms: Public domain W3C validator