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Theorem grothtsk 10827
Description: The Tarski-Grothendieck Axiom, using abbreviations. (Contributed by Mario Carneiro, 28-May-2013.)
Assertion
Ref Expression
grothtsk Tarski = V

Proof of Theorem grothtsk
Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 axgroth5 10816 . . . . 5 𝑥(𝑤𝑥 ∧ ∀𝑦𝑥 (𝒫 𝑦𝑥 ∧ ∃𝑧𝑥 𝒫 𝑦𝑧) ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦𝑥𝑦𝑥))
2 eltskg 10742 . . . . . . . . 9 (𝑥 ∈ V → (𝑥 ∈ Tarski ↔ (∀𝑦𝑥 (𝒫 𝑦𝑥 ∧ ∃𝑧𝑥 𝒫 𝑦𝑧) ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦𝑥𝑦𝑥))))
32elv 3472 . . . . . . . 8 (𝑥 ∈ Tarski ↔ (∀𝑦𝑥 (𝒫 𝑦𝑥 ∧ ∃𝑧𝑥 𝒫 𝑦𝑧) ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦𝑥𝑦𝑥)))
43anbi2i 622 . . . . . . 7 ((𝑤𝑥𝑥 ∈ Tarski) ↔ (𝑤𝑥 ∧ (∀𝑦𝑥 (𝒫 𝑦𝑥 ∧ ∃𝑧𝑥 𝒫 𝑦𝑧) ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦𝑥𝑦𝑥))))
5 3anass 1092 . . . . . . 7 ((𝑤𝑥 ∧ ∀𝑦𝑥 (𝒫 𝑦𝑥 ∧ ∃𝑧𝑥 𝒫 𝑦𝑧) ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦𝑥𝑦𝑥)) ↔ (𝑤𝑥 ∧ (∀𝑦𝑥 (𝒫 𝑦𝑥 ∧ ∃𝑧𝑥 𝒫 𝑦𝑧) ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦𝑥𝑦𝑥))))
64, 5bitr4i 278 . . . . . 6 ((𝑤𝑥𝑥 ∈ Tarski) ↔ (𝑤𝑥 ∧ ∀𝑦𝑥 (𝒫 𝑦𝑥 ∧ ∃𝑧𝑥 𝒫 𝑦𝑧) ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦𝑥𝑦𝑥)))
76exbii 1842 . . . . 5 (∃𝑥(𝑤𝑥𝑥 ∈ Tarski) ↔ ∃𝑥(𝑤𝑥 ∧ ∀𝑦𝑥 (𝒫 𝑦𝑥 ∧ ∃𝑧𝑥 𝒫 𝑦𝑧) ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦𝑥𝑦𝑥)))
81, 7mpbir 230 . . . 4 𝑥(𝑤𝑥𝑥 ∈ Tarski)
9 eluni 4903 . . . 4 (𝑤 Tarski ↔ ∃𝑥(𝑤𝑥𝑥 ∈ Tarski))
108, 9mpbir 230 . . 3 𝑤 Tarski
11 vex 3470 . . 3 𝑤 ∈ V
1210, 112th 264 . 2 (𝑤 Tarski ↔ 𝑤 ∈ V)
1312eqriv 2721 1 Tarski = V
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  wo 844  w3a 1084   = wceq 1533  wex 1773  wcel 2098  wral 3053  wrex 3062  Vcvv 3466  wss 3941  𝒫 cpw 4595   cuni 4900   class class class wbr 5139  cen 8933  Tarskictsk 10740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-groth 10815
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-tsk 10741
This theorem is referenced by:  inaprc  10828  tskmval  10831  tskmcl  10833
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