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Theorem grothtsk 10873
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 10862 . . . . 5 𝑥(𝑤𝑥 ∧ ∀𝑦𝑥 (𝒫 𝑦𝑥 ∧ ∃𝑧𝑥 𝒫 𝑦𝑧) ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦𝑥𝑦𝑥))
2 eltskg 10788 . . . . . . . . 9 (𝑥 ∈ V → (𝑥 ∈ Tarski ↔ (∀𝑦𝑥 (𝒫 𝑦𝑥 ∧ ∃𝑧𝑥 𝒫 𝑦𝑧) ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦𝑥𝑦𝑥))))
32elv 3483 . . . . . . . 8 (𝑥 ∈ Tarski ↔ (∀𝑦𝑥 (𝒫 𝑦𝑥 ∧ ∃𝑧𝑥 𝒫 𝑦𝑧) ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦𝑥𝑦𝑥)))
43anbi2i 623 . . . . . . 7 ((𝑤𝑥𝑥 ∈ Tarski) ↔ (𝑤𝑥 ∧ (∀𝑦𝑥 (𝒫 𝑦𝑥 ∧ ∃𝑧𝑥 𝒫 𝑦𝑧) ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦𝑥𝑦𝑥))))
5 3anass 1094 . . . . . . 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 1845 . . . . 5 (∃𝑥(𝑤𝑥𝑥 ∈ Tarski) ↔ ∃𝑥(𝑤𝑥 ∧ ∀𝑦𝑥 (𝒫 𝑦𝑥 ∧ ∃𝑧𝑥 𝒫 𝑦𝑧) ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦𝑥𝑦𝑥)))
81, 7mpbir 231 . . . 4 𝑥(𝑤𝑥𝑥 ∈ Tarski)
9 eluni 4915 . . . 4 (𝑤 Tarski ↔ ∃𝑥(𝑤𝑥𝑥 ∈ Tarski))
108, 9mpbir 231 . . 3 𝑤 Tarski
11 vex 3482 . . 3 𝑤 ∈ V
1210, 112th 264 . 2 (𝑤 Tarski ↔ 𝑤 ∈ V)
1312eqriv 2732 1 Tarski = V
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wo 847  w3a 1086   = wceq 1537  wex 1776  wcel 2106  wral 3059  wrex 3068  Vcvv 3478  wss 3963  𝒫 cpw 4605   cuni 4912   class class class wbr 5148  cen 8981  Tarskictsk 10786
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-ext 2706  ax-groth 10861
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-tsk 10787
This theorem is referenced by:  inaprc  10874  tskmval  10877  tskmcl  10879
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