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

Theorem grothac 10868
Description: The Tarski-Grothendieck Axiom implies the Axiom of Choice (in the form of cardeqv 10507). This can be put in a more conventional form via ween 10073 and dfac8 10174. 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 10174). (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 4619 . . . . . . . . . 10 (𝑥 = 𝑦 → 𝒫 𝑥 = 𝒫 𝑦)
21sseq1d 4027 . . . . . . . . 9 (𝑥 = 𝑦 → (𝒫 𝑥𝑢 ↔ 𝒫 𝑦𝑢))
31eleq1d 2824 . . . . . . . . 9 (𝑥 = 𝑦 → (𝒫 𝑥𝑢 ↔ 𝒫 𝑦𝑢))
42, 3anbi12d 632 . . . . . . . 8 (𝑥 = 𝑦 → ((𝒫 𝑥𝑢 ∧ 𝒫 𝑥𝑢) ↔ (𝒫 𝑦𝑢 ∧ 𝒫 𝑦𝑢)))
54rspcva 3620 . . . . . . 7 ((𝑦𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ 𝒫 𝑥𝑢)) → (𝒫 𝑦𝑢 ∧ 𝒫 𝑦𝑢))
65simpld 494 . . . . . 6 ((𝑦𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ 𝒫 𝑥𝑢)) → 𝒫 𝑦𝑢)
7 rabss 4082 . . . . . . 7 ({𝑥 ∈ 𝒫 𝑢𝑥𝑢} ⊆ 𝑢 ↔ ∀𝑥 ∈ 𝒫 𝑢(𝑥𝑢𝑥𝑢))
87biimpri 228 . . . . . 6 (∀𝑥 ∈ 𝒫 𝑢(𝑥𝑢𝑥𝑢) → {𝑥 ∈ 𝒫 𝑢𝑥𝑢} ⊆ 𝑢)
9 vex 3482 . . . . . . . . . 10 𝑦 ∈ V
109canth2 9169 . . . . . . . . 9 𝑦 ≺ 𝒫 𝑦
11 sdomdom 9019 . . . . . . . . 9 (𝑦 ≺ 𝒫 𝑦𝑦 ≼ 𝒫 𝑦)
1210, 11ax-mp 5 . . . . . . . 8 𝑦 ≼ 𝒫 𝑦
13 ssdomg 9039 . . . . . . . . 9 (𝑢 ∈ V → (𝒫 𝑦𝑢 → 𝒫 𝑦𝑢))
1413elv 3483 . . . . . . . 8 (𝒫 𝑦𝑢 → 𝒫 𝑦𝑢)
15 domtr 9046 . . . . . . . 8 ((𝑦 ≼ 𝒫 𝑦 ∧ 𝒫 𝑦𝑢) → 𝑦𝑢)
1612, 14, 15sylancr 587 . . . . . . 7 (𝒫 𝑦𝑢𝑦𝑢)
17 vex 3482 . . . . . . . 8 𝑢 ∈ V
18 tskwe 9988 . . . . . . . 8 ((𝑢 ∈ V ∧ {𝑥 ∈ 𝒫 𝑢𝑥𝑢} ⊆ 𝑢) → 𝑢 ∈ dom card)
1917, 18mpan 690 . . . . . . 7 ({𝑥 ∈ 𝒫 𝑢𝑥𝑢} ⊆ 𝑢𝑢 ∈ dom card)
20 numdom 10076 . . . . . . . 8 ((𝑢 ∈ dom card ∧ 𝑦𝑢) → 𝑦 ∈ dom card)
2120expcom 413 . . . . . . 7 (𝑦𝑢 → (𝑢 ∈ dom card → 𝑦 ∈ dom card))
2216, 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 1116 . . . 4 ((𝑦𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ 𝒫 𝑥𝑢) ∧ ∀𝑥 ∈ 𝒫 𝑢(𝑥𝑢𝑥𝑢)) → 𝑦 ∈ dom card)
25 axgroth6 10866 . . . 4 𝑢(𝑦𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ 𝒫 𝑥𝑢) ∧ ∀𝑥 ∈ 𝒫 𝑢(𝑥𝑢𝑥𝑢))
2624, 25exlimiiv 1929 . . 3 𝑦 ∈ dom card
2726, 92th 264 . 2 (𝑦 ∈ dom card ↔ 𝑦 ∈ V)
2827eqriv 2732 1 dom card = V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  {crab 3433  Vcvv 3478  wss 3963  𝒫 cpw 4605   class class class wbr 5148  dom cdm 5689  cdom 8982  csdm 8983  cardccrd 9973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-groth 10861
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-card 9977
This theorem is referenced by:  axgroth3  10869
  Copyright terms: Public domain W3C validator