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Theorem grothtsk 10250
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 10239 . . . . 5 𝑥(𝑤𝑥 ∧ ∀𝑦𝑥 (𝒫 𝑦𝑥 ∧ ∃𝑧𝑥 𝒫 𝑦𝑧) ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦𝑥𝑦𝑥))
2 eltskg 10165 . . . . . . . . 9 (𝑥 ∈ V → (𝑥 ∈ Tarski ↔ (∀𝑦𝑥 (𝒫 𝑦𝑥 ∧ ∃𝑧𝑥 𝒫 𝑦𝑧) ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦𝑥𝑦𝑥))))
32elv 3449 . . . . . . . 8 (𝑥 ∈ Tarski ↔ (∀𝑦𝑥 (𝒫 𝑦𝑥 ∧ ∃𝑧𝑥 𝒫 𝑦𝑧) ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦𝑥𝑦𝑥)))
43anbi2i 625 . . . . . . 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 281 . . . . . 6 ((𝑤𝑥𝑥 ∈ Tarski) ↔ (𝑤𝑥 ∧ ∀𝑦𝑥 (𝒫 𝑦𝑥 ∧ ∃𝑧𝑥 𝒫 𝑦𝑧) ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦𝑥𝑦𝑥)))
76exbii 1849 . . . . 5 (∃𝑥(𝑤𝑥𝑥 ∈ Tarski) ↔ ∃𝑥(𝑤𝑥 ∧ ∀𝑦𝑥 (𝒫 𝑦𝑥 ∧ ∃𝑧𝑥 𝒫 𝑦𝑧) ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦𝑥𝑦𝑥)))
81, 7mpbir 234 . . . 4 𝑥(𝑤𝑥𝑥 ∈ Tarski)
9 eluni 4806 . . . 4 (𝑤 Tarski ↔ ∃𝑥(𝑤𝑥𝑥 ∈ Tarski))
108, 9mpbir 234 . . 3 𝑤 Tarski
11 vex 3447 . . 3 𝑤 ∈ V
1210, 112th 267 . 2 (𝑤 Tarski ↔ 𝑤 ∈ V)
1312eqriv 2798 1 Tarski = V
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wo 844  w3a 1084   = wceq 1538  wex 1781  wcel 2112  wral 3109  wrex 3110  Vcvv 3444  wss 3884  𝒫 cpw 4500   cuni 4803   class class class wbr 5033  cen 8493  Tarskictsk 10163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-groth 10238
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-un 3889  df-in 3891  df-ss 3901  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-tsk 10164
This theorem is referenced by:  inaprc  10251  tskmval  10254  tskmcl  10256
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