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Theorem grothtsk 9849
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 9838 . . . . 5 𝑥(𝑤𝑥 ∧ ∀𝑦𝑥 (𝒫 𝑦𝑥 ∧ ∃𝑧𝑥 𝒫 𝑦𝑧) ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦𝑥𝑦𝑥))
2 vex 3343 . . . . . . . . 9 𝑥 ∈ V
3 eltskg 9764 . . . . . . . . 9 (𝑥 ∈ V → (𝑥 ∈ Tarski ↔ (∀𝑦𝑥 (𝒫 𝑦𝑥 ∧ ∃𝑧𝑥 𝒫 𝑦𝑧) ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦𝑥𝑦𝑥))))
42, 3ax-mp 5 . . . . . . . 8 (𝑥 ∈ Tarski ↔ (∀𝑦𝑥 (𝒫 𝑦𝑥 ∧ ∃𝑧𝑥 𝒫 𝑦𝑧) ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦𝑥𝑦𝑥)))
54anbi2i 732 . . . . . . 7 ((𝑤𝑥𝑥 ∈ Tarski) ↔ (𝑤𝑥 ∧ (∀𝑦𝑥 (𝒫 𝑦𝑥 ∧ ∃𝑧𝑥 𝒫 𝑦𝑧) ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦𝑥𝑦𝑥))))
6 3anass 1081 . . . . . . 7 ((𝑤𝑥 ∧ ∀𝑦𝑥 (𝒫 𝑦𝑥 ∧ ∃𝑧𝑥 𝒫 𝑦𝑧) ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦𝑥𝑦𝑥)) ↔ (𝑤𝑥 ∧ (∀𝑦𝑥 (𝒫 𝑦𝑥 ∧ ∃𝑧𝑥 𝒫 𝑦𝑧) ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦𝑥𝑦𝑥))))
75, 6bitr4i 267 . . . . . 6 ((𝑤𝑥𝑥 ∈ Tarski) ↔ (𝑤𝑥 ∧ ∀𝑦𝑥 (𝒫 𝑦𝑥 ∧ ∃𝑧𝑥 𝒫 𝑦𝑧) ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦𝑥𝑦𝑥)))
87exbii 1923 . . . . 5 (∃𝑥(𝑤𝑥𝑥 ∈ Tarski) ↔ ∃𝑥(𝑤𝑥 ∧ ∀𝑦𝑥 (𝒫 𝑦𝑥 ∧ ∃𝑧𝑥 𝒫 𝑦𝑧) ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦𝑥𝑦𝑥)))
91, 8mpbir 221 . . . 4 𝑥(𝑤𝑥𝑥 ∈ Tarski)
10 eluni 4591 . . . 4 (𝑤 Tarski ↔ ∃𝑥(𝑤𝑥𝑥 ∈ Tarski))
119, 10mpbir 221 . . 3 𝑤 Tarski
12 vex 3343 . . 3 𝑤 ∈ V
1311, 122th 254 . 2 (𝑤 Tarski ↔ 𝑤 ∈ V)
1413eqriv 2757 1 Tarski = V
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 382  wa 383  w3a 1072   = wceq 1632  wex 1853  wcel 2139  wral 3050  wrex 3051  Vcvv 3340  wss 3715  𝒫 cpw 4302   cuni 4588   class class class wbr 4804  cen 8118  Tarskictsk 9762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-groth 9837
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-tsk 9763
This theorem is referenced by:  inaprc  9850  tskmval  9853  tskmcl  9855
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